Discrete Mathematics An Open Introduction Oscar Levin 3rd Edition Discrete Mathematics An Open Introduction Oscar Levin 3rd Edition Oscar Levin School of Mathematical Science University of Northern Colorado Greeley, Co 80639 oscar.levin@unco.edu http://math.oscarlevin.com/ © 2013-2019 by Oscar Levin This work is licensed under the Creative Commons Attribution-ShareAlike 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by-sa/4.0/. 3rd Edition 4th Printing: 12/29/2019 ISBN: 978-1792901690 A current version can always be found for free at http://discrete.openmathbooks.org/ Cover image: Tiling with Fibonacci and Pascal. For Madeline and Teagan Acknowledgements This book would not exist if not for “Discrete and Combinatorial Mathematics” by Richard Grassl and Tabitha Mingus. It is the book I learned discrete math out of, and taught out of the semester before I began writing this text. I wanted to maintain the inquiry based feel of their book but update, expand and rearrange some of the material. Some of the best exposition and exercises here were graciously donated from this source. Thanks to Alees Seehausen who co-taught the Discrete Mathematics course with me in 2015 and helped develop many of the Investigate! activities and other problems currently used in the text. She also oﬀered many suggestions for improvement of the expository text, for which I am quite grateful. Thanks also to Katie Morrison, Nate Eldredge and Richard Grassl (again) for their suggestions after using parts of this text in their classes. While odds are that there are still errors and typos in the current book, there are many fewer thanks to the work of Michelle Morgan over the summer of 2016. The book is now available in an interactive online format, and this is entirely thanks to the work of Rob Beezer, David Farmer, and Alex Jordan along with the rest of the participants of the pretext-support group. Finally, a thank you to the numerous students who have pointed out typos and made suggestions over the years and a thanks in advance to those who will do so in the future. v vi Preface This text aims to give an introduction to select topics in discrete mathematics at a level appropriate for ﬁrst or second year undergraduate math majors, especially those who intend to teach middle and high school mathematics. The book began as a set of notes for the Discrete Mathematics course at the University of Northern Colorado. This course serves both as a survey of the topics in discrete math and as the “bridge” course for math majors, as UNC does not oﬀer a separate “introduction to proofs” course. Most students who take the course plan to teach, although there are a handful of students who will go on to graduate school or study applied math or computer science. For these students the current text hopefully is still of interest, but the intent is not to provide a solid mathematical foundation for computer science, unlike the majority of textbooks on the subject. Another diﬀerence between this text and most other discrete math books is that this book is intended to be used in a class taught using problem oriented or inquiry based methods. When I teach the class, I will assign sections for reading after ﬁrst introducing them in class by using a mix of group work and class discussion on a few interesting problems. The text is meant to consolidate what we discover in class and serve as a reference for students as they master the concepts and techniques covered in the unit. None-the-less, every attempt has been made to make the text suﬃcient for self study as well, in a way that hopefully mimics an inquiry based classroom. The topics covered in this text were chosen to match the needs of the students I teach at UNC. The main areas of study are combinatorics, sequences, logic and proofs, and graph theory, in that order. Induction is covered at the end of the chapter on sequences. Most discrete books put logic ﬁrst as a preliminary, which certainly has its advantages. However, I wanted to discuss logic and proofs together, and found that doing both of these before anything else was overwhelming for my students given that they didn’t yet have context of other problems in the subject. Also, after spending a couple weeks on proofs, we would hardly use that at all when covering combinatorics, so much of the progress we made was quickly lost. Instead, there is a short introduction section on mathematical statements, which should provide enough common language to discuss the logical content of combinatorics and sequences. Depending on the speed of the class, it might be possible to include additional material. In past semesters I have included generating functions (after sequences) and some basic number theory (either after the logic vii viii and proofs chapter or at the very end of the course). These additional topics are covered in the last chapter. While I (currently) believe this selection and order of topics is optimal, you should feel free to skip around to what interests you. There are occasionally examples and exercises that rely on earlier material, but I have tried to keep these to a minimum and usually can either be skipped or understood without too much additional study. If you are an instructor, feel free to edit the LATEX or PreTeXt source to ﬁt your needs. Improvements to the 3rd Edition. In addition to lots of minor corrections, both to typographical and mathematical errors, this third edition includes a few major improvements, including: • More than 100 new exercises, bringing the total to 473. The selection of which exercises have solutions has also been improved, which should make the text more useful for instructors who want to assign homework from the book. • A new section in on trees in the graph theory chapter. • Substantial improvement to the exposition in chapter 0, especially the section on functions. • The interactive online version of the book has added interactivity. Currently, many of the exercises are displayed as WeBWorK problems, allowing readers to enter answers to verify they are correct. The previous editions (2nd edition, released in August 2016, and the Fall 2015 edition) will still be available for instructors who wish to use those versions due to familiarity. My hope is to continue improving the book, releasing a new edition each spring in time for fall adoptions. These new editions will incorporate additions and corrections suggested by instructors and students who use the text the previous semesters. Thus I encourage you to send along any suggestions and comments as you have them. Oscar Levin, Ph.D. University of Northern Colorado, 2019 How to use this book In addition to expository text, this book has a few features designed to encourage you to interact with the mathematics. Investigate! activities. Sprinkled throughout the sections (usually at the very beginning of a topic) you will ﬁnd activities designed to get you acquainted with the topic soon to be discussed. These are similar (sometimes identical) to group activities I give students to introduce material. You really should spend some time thinking about, or even working through, these problems before reading the section. By priming yourself to the types of issues involved in the material you are about to read, you will better understand what is to come. There are no solutions provided for these problems, but don’t worry if you can’t solve them or are not conﬁdent in your answers. My hope is that you will take this frustration with you while you read the proceeding section. By the time you are done with the section, things should be much clearer. Examples. I have tried to include the “correct” number of examples. For those examples which include problems, full solutions are included. Before reading the solution, try to at least have an understanding of what the problem is asking. Unlike some textbooks, the examples are not meant to be all inclusive for problems you will see in the exercises. They should not be used as a blueprint for solving other problems. Instead, use the examples to deepen our understanding of the concepts and techniques discussed in each section. Then use this understanding to solve the exercises at the end of each section. Exercises. You get good at math through practice. Each section concludes with a small number of exercises meant to solidify concepts and basic skills presented in that section. At the end of each chapter, a larger collection of similar exercises is included (as a sort of “chapter review”) which might bridge material of diﬀerent sections in that chapter. Many exercise have a hint or solution (which in the pdf version of the text can be found by clicking on the exercises number—clicking on the solution number will ix x bring you back to the exercise). Readers are encouraged to try these exercises before looking at the help. Both hints and solutions are intended as a way to check your work, but often what would “count” as a correct solution in a math class would be quite a bit more. When I teach with this book, I assign exercises that have solutions as practice and then use them, or similar problems, on quizzes and exams. There are also problems without solutions to challenge yourself (or to be assigned as homework). Interactive Online Version. For those of you reading this in a pdf or in print, I encourage you to also check out the interactive online version, which makes navigating the book a little easier. Additionally, some of the exercises are implemented as WeBWorK problems, which allow you to check your work without seeing the correct answer immediately. Additional interactivity is planned, including instructional videos for examples and additional exercises at the end of sections. These “bonus” features will be added on a rolling basis, so keep an eye out! You can view the interactive version for free at http://discrete.openmathbooks.org/ or by scanning the QR code below with your smart phone. Contents Acknowledgements v Preface vii How to use this book ix 0 Introduction and Preliminaries 0.1 What is Discrete Mathematics? . . . 0.2 Mathematical Statements . . . . . . . Atomic and Molecular Statements . Implications . . . . . . . . . . . . . . Predicates and Quantiﬁers . . . . . . Exercises . . . . . . . . . . . . . . . . 0.3 Sets . . . . . . . . . . . . . . . . . . . Notation . . . . . . . . . . . . . . . . Relationships Between Sets . . . . . Operations On Sets . . . . . . . . . . Venn Diagrams . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . 0.4 Functions . . . . . . . . . . . . . . . . Describing Functions . . . . . . . . . Surjections, Injections, and Bĳections Image and Inverse Image . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 4 4 7 15 17 24 24 28 31 33 35 39 40 45 48 51 . . . . . . . . . . . . . . 57 57 61 64 67 70 70 72 73 74 77 78 81 86 89 . . . . . . . . . . . . . . . . . 1 Counting 1.1 Additive and Multiplicative Principles Counting With Sets . . . . . . . . . . . Principle of Inclusion/Exclusion . . . Exercises . . . . . . . . . . . . . . . . . 1.2 Binomial Coeﬃcients . . . . . . . . . . Subsets . . . . . . . . . . . . . . . . . . Bit Strings . . . . . . . . . . . . . . . . Lattice Paths . . . . . . . . . . . . . . . Binomial Coeﬃcients . . . . . . . . . . Pascal’s Triangle . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . 1.3 Combinations and Permutations . . . Exercises . . . . . . . . . . . . . . . . . 1.4 Combinatorial Proofs . . . . . . . . . . xi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii Contents 1.5 1.6 1.7 Patterns in Pascal’s Triangle . . More Proofs . . . . . . . . . . . Exercises . . . . . . . . . . . . . Stars and Bars . . . . . . . . . . Exercises . . . . . . . . . . . . . Advanced Counting Using PIE Counting Derangements . . . . Counting Functions . . . . . . . Exercises . . . . . . . . . . . . . Chapter Summary . . . . . . . . Chapter Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 95 99 103 108 111 115 117 124 127 128 2 Sequences 2.1 Describing Sequences . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . 2.2 Arithmetic and Geometric Sequences . . . . . . Sums of Arithmetic and Geometric Sequences Exercises . . . . . . . . . . . . . . . . . . . . . . 2.3 Polynomial Fitting . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . 2.4 Solving Recurrence Relations . . . . . . . . . . The Characteristic Root Technique . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . 2.5 Induction . . . . . . . . . . . . . . . . . . . . . . Stamps . . . . . . . . . . . . . . . . . . . . . . . Formalizing Proofs . . . . . . . . . . . . . . . . Examples . . . . . . . . . . . . . . . . . . . . . . Strong Induction . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . 2.6 Chapter Summary . . . . . . . . . . . . . . . . . Chapter Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 136 144 148 151 156 160 164 167 171 175 177 177 179 181 185 188 193 194 3 Symbolic Logic and Proofs 3.1 Propositional Logic . . . . . Truth Tables . . . . . . . . . Logical Equivalence . . . . . Deductions . . . . . . . . . . Beyond Propositions . . . . Exercises . . . . . . . . . . . 3.2 Proofs . . . . . . . . . . . . . Direct Proof . . . . . . . . . Proof by Contrapositive . . Proof by Contradiction . . . Proof by (counter) Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 198 199 201 204 207 209 213 215 216 218 220 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 223 227 228 4 Graph Theory 4.1 Deﬁnitions . . . . . . . . . . . Exercises . . . . . . . . . . . . 4.2 Trees . . . . . . . . . . . . . . Properties of Trees . . . . . . Rooted Trees . . . . . . . . . . Spanning Trees . . . . . . . . Exercises . . . . . . . . . . . . 4.3 Planar Graphs . . . . . . . . . Non-planar Graphs . . . . . . Polyhedra . . . . . . . . . . . Exercises . . . . . . . . . . . . 4.4 Coloring . . . . . . . . . . . . Coloring in General . . . . . . Coloring Edges . . . . . . . . Exercises . . . . . . . . . . . . 4.5 Euler Paths and Circuits . . . Hamilton Paths . . . . . . . . Exercises . . . . . . . . . . . . 4.6 Matching in Bipartite Graphs Exercises . . . . . . . . . . . . 4.7 Chapter Summary . . . . . . . Chapter Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 233 243 247 248 251 253 255 258 260 262 265 267 269 272 274 277 279 280 283 286 289 290 . . . . . . . . . . . . . 295 295 296 299 301 302 304 307 307 310 313 317 319 323 3.3 Proof by Cases . . . Exercises . . . . . . Chapter Summary . Chapter Review . . . . . . . . . . . . . . . . . . . . . . 5 Additional Topics 5.1 Generating Functions . . . . . . . . . . . . . . . . . . . . . Building Generating Functions . . . . . . . . . . . . . . . Diﬀerencing . . . . . . . . . . . . . . . . . . . . . . . . . . Multiplication and Partial Sums . . . . . . . . . . . . . . . Solving Recurrence Relations with Generating Functions Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Introduction to Number Theory . . . . . . . . . . . . . . . Divisibility . . . . . . . . . . . . . . . . . . . . . . . . . . . Remainder Classes . . . . . . . . . . . . . . . . . . . . . . Properties of Congruence . . . . . . . . . . . . . . . . . . Solving Congruences . . . . . . . . . . . . . . . . . . . . . Solving Linear Diophantine Equations . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv Contents A Selected Hints 325 B Selected Solutions 335 C List of Symbols 387 Index 389 Chapter 0 Introduction and Preliminaries Welcome to Discrete Mathematics. If this is your ﬁrst time encountering the subject, you will probably ﬁnd discrete mathematics quite diﬀerent from other math subjects. You might not even know what discrete math is! Hopefully this short introduction will shed some light on what the subject is about and what you can expect as you move forward in your studies. 0.1 What is Discrete Mathematics? dis·crete / dis’krët. Adjective: Individually separate and distinct. Synonyms: separate - detached - distinct - abstract. Deﬁning discrete mathematics is hard because deﬁning mathematics is hard. What is mathematics? The study of numbers? In part, but you also study functions and lines and triangles and parallelepipeds and vectors and . . . . Or perhaps you want to say that mathematics is a collection of tools that allow you to solve problems. What sort of problems? Okay, those that involve numbers, functions, lines, triangles, . . . . Whatever your conception of what mathematics is, try applying the concept of “discrete” to it, as deﬁned above. Some math fundamentally deals with stuﬀ that is individually separate and distinct. In an algebra or calculus class, you might have found a particular set of numbers (maybe the set of numbers in the range of a function). You would represent this set as an interval: [0, ∞) is the range of f (x) x 2 since the set of outputs of the function are all real numbers 0 and greater. This set of numbers is NOT discrete. The numbers in the set are not separated by much at all. In fact, take any two numbers in the set and there are inﬁnitely many more between them which are also in the set. Discrete math could still ask about the range of a function, but the set would not be an interval. Consider the function which gives the number of children of each person reading this. What is the range? I’m guessing it is something like {0, 1, 2, 3}. Maybe 4 is in there too. But certainly there is nobody reading this that has 1.32419 children. This output set is discrete because the elements are separate. The inputs to the function also form a discrete set because each input is an individual person. One way to get a feel for the subject is to consider the types of problems you solve in discrete math. Here are a few simple examples: 1 2 0. Introduction and Preliminaries Investigate! Note: Throughout the text you will see Investigate! activities like this one. Answer the questions in these as best you can to give yourself a feel for what is coming next. 1. The most popular mathematician in the world is throwing a party for all of his friends. As a way to kick things oﬀ, they decide that everyone should shake hands. Assuming all 10 people at the party each shake hands with every other person (but not themselves, obviously) exactly once, how many handshakes take place? 2. At the warm-up event for Oscar’s All Star Hot Dog Eating Contest, Al ate one hot dog. Bob then showed him up by eating three hot dogs. Not to be outdone, Carl ate ﬁve. This continued with each contestant eating two more hot dogs than the previous contestant. How many hot dogs did Zeno (the 26th and ﬁnal contestant) eat? How many hot dogs were eaten all together? 3. After excavating for weeks, you ﬁnally arrive at the burial chamber. The room is empty except for two large chests. On each is carved a message (strangely in English): If this chest is empty, then the other chest’s message is true. This chest is ﬁlled with treasure or the other chest contains deadly scorpions. You know exactly one of these messages is true. What should you do? 4. Back in the days of yore, ﬁve small towns decided they wanted to build roads directly connecting each pair of towns. While the towns had plenty of money to build roads as long and as winding as they wished, it was very important that the roads not intersect with each other (as stop signs had not yet been invented). Also, tunnels and bridges were not allowed. Is it possible for each of these towns to build a road to each of the four other towns without creating any intersections? ! Attempt the above activity before proceeding ! One reason it is diﬃcult to deﬁne discrete math is that it is a very broad description which encapsulates a large number of subjects. In 0.1. What is Discrete Mathematics? 3 this course we will study four main topics: combinatorics (the theory of ways things combine; in particular, how to count these ways), sequences, symbolic logic, and graph theory. However, there are other topics that belong under the discrete umbrella, including computer science, abstract algebra, number theory, game theory, probability, and geometry (some of these, particularly the last two, have both discrete and non-discrete variants). Ultimately the best way to learn what discrete math is about is to do it. Let’s get started! Before we can begin answering more complicated (and fun) problems, we must lay down some foundation. We start by reviewing mathematical statements, sets, and functions in the framework of discrete mathematics. 4 0. Introduction and Preliminaries 0.2 Mathematical Statements Investigate! While walking through a ﬁctional forest, you encounter three trolls guarding a bridge. Each is either a knight, who always tells the truth, or a knave, who always lies. The trolls will not let you pass until you correctly identify each as either a knight or a knave. Each troll makes a single statement: Troll 1: If I am a knave, then there are exactly two knights here. Troll 2: Troll 1 is lying. Troll 3: Either we are all knaves or at least one of us is a knight. Which troll is which? ! Attempt the above activity before proceeding ! In order to do mathematics, we must be able to talk and write about mathematics. Perhaps your experience with mathematics so far has mostly involved ﬁnding answers to problems. As we embark towards more advanced and abstract mathematics, writing will play a more prominent role in the mathematical process. Communication in mathematics requires more precision than many other subjects, and thus we should take a few pages here to consider the basic building blocks: mathematical statements. Atomic and Molecular Statements A statement is any declarative sentence which is either true or false. A statement is atomic if it cannot be divided into smaller statements, otherwise it is called molecular. Example 0.2.1 These are statements (in fact, atomic statements): • Telephone numbers in the USA have 10 digits. • The moon is made of cheese. • 42 is a perfect square. • Every even number greater than 2 can be expressed as the sum of two primes. 0.2. Mathematical Statements 5 • 3 + 7 12 And these are not statements: • Would you like some cake? • The sum of two squares. • 1 + 3 + 5 + 7 + · · · + 2n + 1. • Go to your room! • 3 + x 12 The reason the sentence “3 + x 12” is not a statement is that it contains a variable. Depending on what x is, the sentence is either true or false, but right now it is neither. One way to make the sentence into a statement is to specify the value of the variable in some way. This could be done by specifying a speciﬁc substitution, for example, “3 + x 12 where x 9,” which is a true statement. Or you could capture the free variable by quantifying over it, as in, “for all values of x, 3 + x 12,” which is false. We will discuss quantiﬁers in more detail at the end of this section. You can build more complicated (molecular) statements out of simpler (atomic or molecular) ones using logical connectives. For example, this is a molecular statement: Telephone numbers in the USA have 10 digits and 42 is a perfect square. Note that we can break this down into two smaller statements. The two shorter statements are connected by an “and.” We will consider 5 connectives: “and” (Sam is a man and Chris is a woman), “or” (Sam is a man or Chris is a woman), “if. . . , then. . . ” (if Sam is a man, then Chris is a woman), “if and only if” (Sam is a man if and only if Chris is a woman), and “not” (Sam is not a man). The ﬁrst four are called binary connectives (because they connect two statements) while “not” is an example of a unary connective (since it applies to a single statement). These molecular statements are of course still statements, so they must be either true or false. The absolutely key observation here is that which truth value the molecular statement achieves is completely determined by the type of connective and the truth values of the parts. We do not need to know what the parts actually say, only whether those parts are true or false. So to analyze logical connectives, it is enough to consider propositional variables (sometimes called sentential variables), usually capital letters in the middle of the alphabet: P, Q, R, S, . . .. We think of these as standing in for (usually atomic) statements, but there are only 6 0. Introduction and Preliminaries two values the variables can achieve: true or false.1 We also have symbols for the logical connectives: ∧, ∨, →, ↔, ¬. Logical Connectives. • P ∧ Q is read “P and Q,” and called a conjunction. • P ∨ Q is read “P or Q,” and called a disjunction. • P → Q is read “if P then Q,” and called an implication or conditional. • P ↔ Q is read “P if and only if Q,” and called a biconditional. • ¬P is read “not P,” and called a negation. The truth value of a statement is determined by the truth value(s) of its part(s), depending on the connectives: Truth Conditions for Connectives. • P ∧ Q is true when both P and Q are true • P ∨ Q is true when P or Q or both are true. • P → Q is true when P is false or Q is true or both. • P ↔ Q is true when P and Q are both true, or both false. • ¬P is true when P is false. Note that for us, or is the inclusive or (and not the sometimes used exclusive or) meaning that P ∨ Q is in fact true when both P and Q are true. As for the other connectives, “and” behaves as you would expect, as does negation. The biconditional (if and only if) might seem a little strange, but you should think of this as saying the two parts of the statements are equivalent in that they have the same truth value. This leaves only the conditional P → Q which has a slightly diﬀerent meaning in mathematics than it does in ordinary usage. However, implications are so common and useful in mathematics, that we must develop ﬂuency with their use, and as such, they deserve their own subsection. 1In computer programing, we should call such variables Boolean variables. 0.2. Mathematical Statements 7 Implications Implications. An implication or conditional is a molecular statement of the form P→Q where P and Q are statements. We say that • P is the hypothesis (or antecedent). • Q is the conclusion (or consequent). An implication is true provided P is false or Q is true (or both), and false otherwise. In particular, the only way for P → Q to be false is for P to be true and Q to be false. Easily the most common type of statement in mathematics is the implication. Even statements that do not at ﬁrst look like they have this form conceal an implication at their heart. Consider the Pythagorean Theorem. Many a college freshman would quote this theorem as “a 2 + b 2 c 2 .” This is absolutely not correct. For one thing, that is not a statement since it has three variables in it. Perhaps they imply that this should be true for any values of the variables? So 12 + 52 22 ??? How can we ﬁx this? Well, the equation is true as long as a and b are the legs or a right triangle and c is the hypotenuse. In other words: If a and b are the legs of a right triangle with hypotenuse c, then a 2 + b 2 c 2 . This is a reasonable way to think about implications: our claim is that the conclusion (“then” part) is true, but on the assumption that the hypothesis (“if” part) is true. We make no claim about the conclusion in situations when the hypothesis is false.2 Still, it is important to remember that an implication is a statement, and therefore is either true or false. The truth value of the implication is determined by the truth values of its two parts. To agree with the usage above, we say that an implication is true either when the hypothesis is false, or when the conclusion is true. This leaves only one way for an implication to be false: when the hypothesis is true and the conclusion is false. 2However, note that in the case of the Pythagorean Theorem, it is also the case that if a 2 + b 2 c 2 , then a and b are the legs of a right triangle with hypotenuse c. So we could have also expressed this theorem as a biconditional: “a and b are the legs of a right triangle with hypotenuse c if and only if a 2 + b 2 c 2 .” 8 0. Introduction and Preliminaries Example 0.2.2 Consider the statement: If Bob gets a 90 on the ﬁnal, then Bob will pass the class. This is deﬁnitely an implication: P is the statement “Bob gets a 90 on the ﬁnal,” and Q is the statement “Bob will pass the class.” Suppose I made that statement to Bob. In what circumstances would it be fair to call me a liar? What if Bob really did get a 90 on the ﬁnal, and he did pass the class? Then I have not lied; my statement is true. However, if Bob did get a 90 on the ﬁnal and did not pass the class, then I lied, making the statement false. The tricky case is this: what if Bob did not get a 90 on the ﬁnal? Maybe he passes the class, maybe he doesn’t. Did I lie in either case? I think not. In these last two cases, P was false, and the statement P → Q was true. In the ﬁrst case, Q was true, and so was P → Q. So P → Q is true when either P is false or Q is true. Just to be clear, although we sometimes read P → Q as “P implies Q”, we are not insisting that there is some causal relationship between the statements P and Q. In particular, if you claim that P → Q is false, you are not saying that P does not imply Q, but rather that P is true and Q is false. Example 0.2.3 Decide which of the following statements are true and which are false. Brieﬂy explain. 1. If 1 1, then most horses have 4 legs. 2. If 0 1, then 1 1. 3. If 8 is a prime number, then the 7624th digit of π is an 8. 4. If the 7624th digit of π is an 8, then 2 + 2 4. Solution. All four of the statements are true. Remember, the only way for an implication to be false is for the if part to be true and the then part to be false. 1. Here both the hypothesis and the conclusion are true, so the implication is true. It does not matter that there is no meaningful connection between the true mathematical fact and the fact about horses. 2. Here the hypothesis is false and the conclusion is true, so the implication is true. 0.2. Mathematical Statements 9 3. I have no idea what the 7624th digit of π is, but this does not matter. Since the hypothesis is false, the implication is automatically true. 4. Similarly here, regardless of the truth value of the hypothesis, the conclusion is true, making the implication true. It is important to understand the conditions under which an implication is true not only to decide whether a mathematical statement is true, but in order to prove that it is. Proofs might seem scary (especially if you have had a bad high school geometry experience) but all we are really doing is explaining (very carefully) why a statement is true. If you understand the truth conditions for an implication, you already have the outline for a proof. Direct Proofs of Implications. To prove an implication P → Q, it is enough to assume P, and from it, deduce Q. Perhaps a better way to say this is that to prove a statement of the form P → Q directly, you must explain why Q is true, but you get to assume P is true ﬁrst. After all, you only care about whether Q is true in the case that P is as well. There are other techniques to prove statements (implications and others) that we will encounter throughout our studies, and new proof techniques are discovered all the time. Direct proof is the easiest and most elegant style of proof and has the advantage that such a proof often does a great job of explaining why the statement is true. Example 0.2.4 Prove: If two numbers a and b are even, then their sum a + b is even. Solution. Proof. Suppose the numbers a and b are even. This means that a 2k and b 2j for some integers k and j. The sum is then a + b 2k + 2j 2(k + j). Since k + j is an integer, this means that a + b is even. qed Notice that since we get to assume the hypothesis of the implication, we immediately have a place to start. The proof proceeds essentially by repeatedly asking and answering, “what does that mean?” Eventually, we conclude that it means the conclusion. 10 0. Introduction and Preliminaries This sort of argument shows up outside of math as well. If you ever found yourself starting an argument with “hypothetically, let’s assume . . . ,” then you have attempted a direct proof of your desired conclusion. An implication is a way of expressing a relationship between two statements. It is often interesting to ask whether there are other relationships between the statements. Here we introduce some common language to address this question. Converse and Contrapositive. • The converse of an implication P → Q is the implication Q → P. The converse is NOT logically equivalent to the original implication. That is, whether the converse of an implication is true is independent of the truth of the implication. • The contrapositive of an implication P → Q is the statement ¬Q → ¬P. An implication and its contrapositive are logically equivalent (they are either both true or both false). Mathematics is overﬂowing with examples of true implications which have a false converse. If a number greater than 2 is prime, then that number is odd. However, just because a number is odd does not mean it is prime. If a shape is a square, then it is a rectangle. But it is false that if a shape is a rectangle, then it is a square. However, sometimes the converse of a true statement is also true. For example, the Pythagorean theorem has a true converse: ifa 2 + b 2 c 2 , then the triangle with sides a, b, and c is a right triangle. Whenever you encounter an implication in mathematics, it is always reasonable to ask whether the converse is true. The contrapositive, on the other hand, always has the same truth value as its original implication. This can be very helpful in deciding whether an implication is true: often it is easier to analyze the contrapositive. Example 0.2.5 True or false: If you draw any nine playing cards from a regular deck, then you will have at least three cards all of the same suit. Is the converse true? Solution. True. The original implication is a little hard to analyze because there are so many diﬀerent combinations of nine cards. But consider the contrapositive: If you don’t have at least three cards all of the same suit, then you don’t have nine cards. It is easy to see 0.2. Mathematical Statements 11 why this is true: you can at most have two cards of each of the four suits, for a total of eight cards (or fewer). The converse: If you have at least three cards all of the same suit, then you have nine cards. This is false. You could have three spades and nothing else. Note that to demonstrate that the converse (an implication) is false, we provided an example where the hypothesis is true (you do have three cards of the same suit), but where the conclusion is false (you do not have nine cards). Understanding converses and contrapositives can help understand implications and their truth values: Example 0.2.6 Suppose I tell Sue that if she gets a 93% on her ﬁnal, then she will get an A in the class. Assuming that what I said is true, what can you conclude in the following cases: 1. Sue gets a 93% on her ﬁnal. 2. Sue gets an A in the class. 3. Sue does not get a 93% on her ﬁnal. 4. Sue does not get an A in the class. Solution. Note ﬁrst that whenever P → Q and P are both true statements, Q must be true as well. For this problem, take P to mean “Sue gets a 93% on her ﬁnal” and Q to mean “Sue will get an A in the class.” 1. We have P → Q and P, so Q follows. Sue gets an A. 2. You cannot conclude anything. Sue could have gotten the A because she did extra credit for example. Notice that we do not know that if Sue gets an A, then she gets a 93% on her ﬁnal. That is the converse of the original implication, so it might or might not be true. 3. The contrapositive of the converse of P → Q is ¬P → ¬Q, which states that if Sue does not get a 93% on the ﬁnal, then she will not get an A in the class. But this does not follow from the original implication. Again, we can conclude nothing. Sue could have done extra credit. 12 0. Introduction and Preliminaries 4. What would happen if Sue does not get an A but did get a 93% on the ﬁnal? Then P would be true and Q would be false. This makes the implication P → Q false! It must be that Sue did not get a 93% on the ﬁnal. Notice now we have the implication ¬Q → ¬P which is the contrapositive of P → Q. Since P → Q is assumed to be true, we know ¬Q → ¬P is true as well. As we said above, an implication is not logically equivalent to its converse, but it is possible that both the implication and its converse are true. In this case, when both P → Q and Q → P are true, we say that P and Q are equivalent and write P ↔ Q. This is the biconditional we mentioned earlier. If and only if. P ↔ Q is logically equivalent to (P → Q) ∧ (Q → P). Example: Given an integer n, it is true that n is even if and only if n 2 is even. That is, if n is even, then n 2 is even, as well as the converse: if n 2 is even, then n is even. You can think of “if and only if” statements as having two parts: an implication and its converse. We might say one is the “if” part, and the other is the “only if” part. We also sometimes say that “if and only if” statements have two directions: a forward direction (P → Q) and a backwards direction (P ← Q, which is really just sloppy notation for Q → P). Let’s think a little about which part is which. Is P → Q the “if” part or the “only if” part? Consider an example. Example 0.2.7 Suppose it is true that I sing if and only if I’m in the shower. We know this means both that if I sing, then I’m in the shower, and also the converse, that if I’m in the shower, then I sing. Let P be the statement, “I sing,” and Q be, “I’m in the shower.” So P → Q is the statement “if I sing, then I’m in the shower.” Which part of the if and only if statement is this? What we are really asking for is the meaning of “I sing if I’m in the shower” and “I sing only if I’m in the shower.” When is the ﬁrst one (the “if” part) false? When I am in the shower but not singing. That is the same condition on being false as the statement “if I’m in the shower, then I sing.” So the “if” part is Q → P. On the 0.2. Mathematical Statements 13 other hand, to say, “I sing only if I’m in the shower” is equivalent to saying “if I sing, then I’m in the shower,” so the “only if” part is P → Q. It is not terribly important to know which part is the “if” or “only if” part, but this does illustrate something very, very important: there are many ways to state an implication! Example 0.2.8 Rephrase the implication, “if I dream, then I am asleep” in as many diﬀerent ways as possible. Then do the same for the converse. Solution. The following are all equivalent to the original implication: 1. I am asleep if I dream. 2. I dream only if I am asleep. 3. In order to dream, I must be asleep. 4. To dream, it is necessary that I am asleep. 5. To be asleep, it is suﬃcient to dream. 6. I am not dreaming unless I am asleep. The following are equivalent to the converse (if I am asleep, then I dream): 1. I dream if I am asleep. 2. I am asleep only if I dream. 3. It is necessary that I dream in order to be asleep. 4. It is suﬃcient that I be asleep in order to dream. 5. If I don’t dream, then I’m not asleep. Hopefully you agree with the above example. We include the “necessary and suﬃcient” versions because those are common when discussing mathematics. In fact, let’s agree once and for all what they mean. 14 0. Introduction and Preliminaries Necessary and Suﬃcient. • “P is necessary for Q” means Q → P. • “P is suﬃcient for Q” means P → Q. • If P is necessary and suﬃcient for Q, then P ↔ Q. To be honest, I have trouble with these if I’m not very careful. I ﬁnd it helps to keep a standard example for reference. Example 0.2.9 Recall from calculus, if a function is diﬀerentiable at a point c, then it is continuous at c, but that the converse of this statement is not true (for example, f (x) |x| at the point 0). Restate this fact using “necessary and suﬃcient” language. Solution. It is true that in order for a function to be diﬀerentiable at a point c, it is necessary for the function to be continuous at c. However, it is not necessary that a function be diﬀerentiable at c for it to be continuous at c. It is true that to be continuous at a point c, it is suﬃcient that the function be diﬀerentiable at c. However, it is not the case that being continuous at c is suﬃcient for a function to be diﬀerentiable at c. Thinking about the necessity and suﬃciency of conditions can also help when writing proofs and justifying conclusions. If you want to establish some mathematical fact, it is helpful to think what other facts would be enough (be suﬃcient) to prove your fact. If you have an assumption, think about what must also be necessary if that hypothesis is true. 0.2. Mathematical Statements 15 Predicates and Quantifiers Investigate! Consider the statements below. Decide whether any are equivalent to each other, or whether any imply any others. 1. You can fool some people all of the time. 2. You can fool everyone some of the time. 3. You can always fool some people. 4. Sometimes you can fool everyone. ! Attempt the above activity before proceeding ! It would be nice to use variables in our mathematical sentences. For example, suppose we wanted to claim that if n is prime, then n + 7 is not prime. This looks like an implication. I would like to write something like P(n) → ¬P(n + 7) where P(n) means “n is prime.” But this is not quite right. For one thing, because this sentence has a free variable (that is, a variable that we have not speciﬁed anything about), it is not a statement. A sentence that contains variables is called a predicate. Now, if we plug in a speciﬁc value for n, we do get a statement. In fact, it turns out that no matter what value we plug in for n, we get a true implication in this case. What we really want to say is that for all values of n, if n is prime, then n + 7 is not. We need to quantify the variable. Although there are many types of quantiﬁers in English (e.g., many, few, most, etc.) in mathematics we, for the most part, stick to two: existential and universal. Universal and Existential Quantiﬁers. The existential quantiﬁer is ∃ and is read “there exists” or “there is.” For example, ∃x(x < 0) asserts that there is a number less than 0. The universal quantiﬁer is ∀ and is read “for all” or “every.” For example, ∀x(x ≥ 0) asserts that every number is greater than or equal to 0. 16 0. Introduction and Preliminaries As with all mathematical statements, we would like to decide whether quantiﬁed statements are true or false. Consider the statement ∀x∃y(y < x). You would read this, “for every x there is some y such that y is less than x.” Is this true? The answer depends on what our domain of discourse is: when we say “for all” x, do we mean all positive integers or all real numbers or all elements of some other set? Usually this information is implied. In discrete mathematics, we almost always quantify over the natural numbers, 0, 1, 2, . . . , so let’s take that for our domain of discourse here. For the statement to be true, we need it to be the case that no matter what natural number we select, there is always some natural number that is strictly smaller. Perhaps we could let y be x −1? But here is the problem: what if x 0? Then y −1 and that is not a number! (in our domain of discourse). Thus we see that the statement is false because there is a number which is less than or equal to all other numbers. In symbols, ∃x∀y(y ≥ x). To show that the original statement is false, we proved that the negation was true. Notice how the negation and original statement compare. This is typical. Quantiﬁers and Negation. ¬∀xP(x) is equivalent to ∃x¬P(x). ¬∃xP(x) is equivalent to ∀x¬P(x). Essentially, we can pass the negation symbol over a quantiﬁer, but that causes the quantiﬁer to switch type. This should not be surprising: if not everything has a property, then something doesn’t have that property. And if there is not something with a property, then everything doesn’t have that property. Implicit Quantifiers. It is always a good idea to be precise in mathematics. Sometimes though, we can relax a little bit, as long as we all agree on a convention. An example of such a convention is to assume that sentences containing predicates with free variables are intended as statements, where the variables are universally quantiﬁed. For example, do you believe that if a shape is a square, then it is a rectangle? But how can that be true if it is not a statement? To be a little 0.2. Mathematical Statements 17 more precise, we have two predicates: S(x) standing for “x is a square” and R(x) standing for “x is a rectangle”. The sentence we are looking at is, S(x) → R(x). This is neither true nor false, as it is not a statement. But come on! We all know that we meant to consider the statement, ∀x(S(x) → R(x)), and this is what our convention tells us to consider. Similarly, we will often be a bit sloppy about the distinction between a predicate and a statement. For example, we might write, let P(n) be the statement, “n is prime,” which is technically incorrect. It is implicit that we mean that we are deﬁning P(n) to be a predicate, which for each n becomes the statement, n is prime. Exercises 1. For each sentence below, decide whether it is an atomic statement, a molecular statement, or not a statement at all. (a) Customers must wear shoes. (b) The customers wore shoes. (c) The customers wore shoes and they wore socks. 2. Classify each of the sentences below as an atomic statement, a molecular statement, or not a statement at all. If the statement is molecular, say what kind it is (conjuction, disjunction, conditional, biconditional, negation). (a) The sum of the ﬁrst 100 odd positive integers. (b) Everybody needs somebody sometime. (c) The Broncos will win the Super Bowl or I’ll eat my hat. (d) We can have donuts for dinner, but only if it rains. (e) Every natural number greater than 1 is either prime or composite. (f) This sentence is false. 3. Suppose P and Q are the statements: P: Jack passed math. Q: Jill passed math. (a) Translate “Jack and Jill both passed math” into symbols. (b) Translate “If Jack passed math, then Jill did not” into symbols. (c) Translate “P ∨ Q” into English. 18 0. Introduction and Preliminaries (d) Translate “¬(P ∧ Q) → Q” into English. (e) Suppose you know that if Jack passed math, then so did Jill. What can you conclude if you know that: i. Jill passed math? ii. Jill did not pass math? 4. Determine whether each molecular statement below is true or false, or whether it is impossible to determine. Assume you do not know what my favorite number is (but you do know that 13 is prime). (a) If 13 is prime, then 13 is my favorite number. (b) If 13 is my favorite number, then 13 is prime. (c) If 13 is not prime, then 13 is my favorite number. (d) 13 is my favorite number or 13 is prime. (e) 13 is my favorite number and 13 is prime. (f) 7 is my favorite number and 13 is not prime. (g) 13 is my favorite number or 13 is not my favorite number. 5. In my safe is a sheet of paper with two shapes drawn on it in colored crayon. One is a square, and the other is a triangle. Each shape is drawn in a single color. Suppose you believe me when I tell you that if the square is blue, then the triangle is green. What do you therefore know about the truth value of the following statements? (a) The square and the triangle are both blue. (b) The square and the triangle are both green. (c) If the triangle is not green, then the square is not blue. (d) If the triangle is green, then the square is blue. (e) The square is not blue or the triangle is green. 6. Again, suppose the statement “if the square is blue, then the triangle is green” is true. This time however, assume the converse is false. Classify each statement below as true or false (if possible). (a) The square is blue if and only if the triangle is green. (b) The square is blue if and only if the triangle is not green. (c) The square is blue. (d) The triangle is green. 0.2. Mathematical Statements 7. 19 Consider the statement, “If you will give me a cow, then I will give you magic beans.” Decide whether each statement below is the converse, the contrapositive, or neither. (a) If you will give me a cow, then I will not give you magic beans. (b) If I will not give you magic beans, then you will not give me a cow. (c) If I will give you magic beans, then you will give me a cow. (d) If you will not give me a cow, then I will not give you magic beans. (e) You will give me a cow and I will not give you magic beans. (f) If I will give you magic beans, then you will not give me a cow. 8. Consider the statement “If Oscar eats Chinese food, then he drinks milk.” (a) Write the converse of the statement. (b) Write the contrapositive of the statement. (c) Is it possible for the contrapositive to be false? If it was, what would that tell you? (d) Suppose the original statement is true, and that Oscar drinks milk. Can you conclude anything (about his eating Chinese food)? Explain. (e) Suppose the original statement is true, and that Oscar does not drink milk. Can you conclude anything (about his eating Chinese food)? Explain. 9. You have discovered an old paper on graph theory that discusses the viscosity of a graph (which for all you know, is something completely made up by the author). A theorem in the paper claims that “if a graph satisﬁes condition (V), then the graph is viscous.” Which of the following are equivalent ways of stating this claim? Which are equivalent to the converse of the claim? (a) A graph is viscous only if it satisﬁes condition (V). (b) A graph is viscous if it satisﬁes condition (V). (c) For a graph to be viscous, it is necessary that it satisﬁes condition (V). (d) For a graph to be viscous, it is suﬃcient for it to satisfy condition (V). 20 0. Introduction and Preliminaries (e) Satisfying condition (V) is a suﬃcient condition for a graph to be viscous. (f) Satisfying condition (V) is a necessary condition for a graph to be viscous. (g) Every viscous graph satisﬁes condition (V). (h) Only viscous graphs satisfy condition (V). 10. Write each of the following statements in the form, “if . . . , then . . . .” Careful, some of the statements might be false (which is alright for the purposes of this question). (a) To lose weight, you must exercise. (b) To lose weight, all you need to do is exercise. (c) Every American is patriotic. (d) You are patriotic only if you are American. (e) The set of rational numbers is a subset of the real numbers. (f) A number is prime if it is not even. (g) Either the Broncos will win the Super Bowl, or they won’t play in the Super Bowl. 11. Which of the following statements are equivalent to the implication, “if you win the lottery, then you will be rich,” and which are equivalent to the converse of the implication? (a) Either you win the lottery or else you are not rich. (b) Either you don’t win the lottery or else you are rich. (c) You will win the lottery and be rich. (d) You will be rich if you win the lottery. (e) You will win the lottery if you are rich. (f) It is necessary for you to win the lottery to be rich. (g) It is suﬃcient to win the lottery to be rich. (h) You will be rich only if you win the lottery. (i) Unless you win the lottery, you won’t be rich. (j) If you are rich, you must have won the lottery. (k) If you are not rich, then you did not win the lottery. (l) You will win the lottery if and only if you are rich. 0.2. Mathematical Statements 21 12. Let P(x) be the predicate, “3x + 1 is even.” (a) Is P(5) true or false? (b) What, if anything, can you conclude about ∃xP(x) from the truth value of P(5)? (c) What, if anything, can you conclude about ∀xP(x) from the truth value of P(5)? 13. Let P(x) be the predicate, “4x + 1 is even.” (a) Is P(5) true or false? (b) What, if anything, can you conclude about ∃xP(x) from the truth value of P(5)? (c) What, if anything, can you conclude about ∀xP(x) from the truth value of P(5)? 14. For a given predicate P(x), you might believe that the statements ∀xP(x) or ∃xP(x) are either true or false. How would you decide if you were correct in each case? You have four choices: you could give an example of an element n in the domain for which P(n) is true or for which P(n) if false, or you could argue that no matter what n is, P(n) is true or is false. (a) What would you need to do to prove ∀xP(x) is true? (b) What would you need to do to prove ∀xP(x) is false? (c) What would you need to do to prove ∃xP(x) is true? (d) What would you need to do to prove ∃xP(x) is false? 15. Suppose P(x, y) is some binary predicate deﬁned on a very small domain of discourse: just the integers 1, 2, 3, and 4. For each of the 16 pairs of these numbers, P(x, y) is either true or false, according to the following table (x values are rows, y values are columns). 1 2 3 4 1 T F F F 2 F T T F 3 T T T T 4 F F F F For example, P(1, 3) is false, as indicated by the F in the ﬁrst row, third column. Use the table to decide whether the following statements are true or false. (a) ∀x∃yP(x, y). 22 0. Introduction and Preliminaries (b) ∀y∃xP(x, y). (c) ∃x∀yP(x, y). (d) ∃y∀xP(x, y). 16. Translate into symbols. Use E(x) for “x is even” and O(x) for “x is odd.” (a) No number is both even and odd. (b) One more than any even number is an odd number. (c) There is prime number that is even. (d) Between any two numbers there is a third number. (e) There is no number between a number and one more than that number. 17. Translate into English: (a) ∀x(E(x) → E(x + 2)). (b) ∀x∃y(sin(x) y). (c) ∀y∃x(sin(x) y). (d) ∀x∀y(x 3 y 3 → x y). 18. Suppose P(x) is some predicate for which the statement ∀xP(x) is true. Is it also the case that ∃xP(x) is true? In other words, is the statement ∀xP(x) → ∃xP(x) always true? Is the converse always true? Assume the domain of discourse is non-empty. 19. For each of the statements below, give a domain of discourse for which the statement is true, and a domain for which the statement is false. (a) ∀x∃y(y 2 x). (b) ∀x∀y(x < y → ∃z(x < z < y)). (c) ∃x∀y∀z(y < z → y ≤ x ≤ z). 20. Consider the statement, “For all natural numbers n, if n is prime, then n is solitary.” You do not need to know what solitary means for this problem, just that it is a property that some numbers have and others do not. (a) Write the converse and the contrapositive of the statement, saying which is which. Note: the original statement claims that an implication is true for all n, and it is that implication that we are taking the converse and contrapositive of. 0.2. Mathematical Statements 23 (b) Write the negation of the original statement. What would you need to show to prove that the statement is false? (c) Even though you don’t know whether 10 is solitary (in fact, nobody knows this), is the statement “if 10 is prime, then 10 is solitary” true or false? Explain. (d) It turns out that 8 is solitary. Does this tell you anything about the truth or falsity of the original statement, its converse or its contrapositive? Explain. (e) Assuming that the original statement is true, what can you say about the relationship between the set P of prime numbers and the set S of solitary numbers. Explain. 24 0. Introduction and Preliminaries 0.3 Sets The most fundamental objects we will use in our studies (and really in all of math) are sets. Much of what follows might be review, but it is very important that you are ﬂuent in the language of set theory. Most of the notation we use below is standard, although some might be a little diﬀerent than what you have seen before. For us, a set will simply be an unordered collection of objects. Two examples: we could consider the set of all actors who have played The Doctor on Doctor Who, or the set of natural numbers between 1 and 10 inclusive. In the ﬁrst case, Tom Baker is a element (or member) of the set, while Idris Elba, among many others, is not an element of the set. Also, the two examples are of diﬀerent sets. Two sets are equal exactly if they contain the exact same elements. For example, the set containing all of the vowels in the declaration of independence is precisely the same set as the set of vowels in the word “questionably” (namely, all of them); we do not care about order or repetitions, just whether the element is in the set or not. Notation We need some notation to make talking about sets easier. Consider, A {1, 2, 3}. This is read, “A is the set containing the elements 1, 2 and 3.” We use curly braces “{, }” to enclose elements of a set. Some more notation: a ∈ {a, b, c}. The symbol “∈” is read “is in” or “is an element of.” Thus the above means that a is an element of the set containing the letters a, b, and c. Note that this is a true statement. It would also be true to say that d is not in that set: d < {a, b, c}. Be warned: we write “x ∈ A” when we wish to express that one of the elements of the set A is x. For example, consider the set, A {1, b, {x, y, z}, ∅}. This is a strange set, to be sure. It contains four elements: the number 1, the letter b, the set {x, y, z}, and the empty set ∅ {}, the set containing no elements. Is x in A? The answer is no. None of the four elements in A are the letter x, so we must conclude that x < A. Similarly, consider the set B {1, b}. Even though the elements of B are elements of A, we 0.3. Sets 25 cannot say that the set B is one of the elements of A. Therefore B < A. (Soon we will see that B is a subset of A, but this is diﬀerent from being an element of A.) We have described the sets above by listing their elements. Sometimes this is hard to do, especially when there are a lot of elements in the set (perhaps inﬁnitely many). For instance, if we want A to be the set of all even natural numbers, would could write, A {0, 2, 4, 6, . . .}, but this is a little imprecise. A better way would be A {x ∈ N : ∃n ∈ N(x 2n)}. Let’s look at this carefully. First, there are some new symbols to digest: “N” is the symbol usually used to denote that natural numbers, which we will take to be the set {0, 1, 2, 3, . . .}. Next, the colon, “:”, is read such that; it separates the elements that are in the set from the condition that the elements in the set must satisfy. So putting this all together, we would read the set as, “the set of all x in the natural numbers, such that there exists some n in the natural numbers for which x is twice n.” In other words, the set of all natural numbers, that are even. Here is another way to write the same set. A {x ∈ N : x is even}. Note: Sometimes mathematicians use | or ϶ for the “such that” symbol instead of the colon. Also, there is a fairly even split between mathematicians about whether 0 is an element of the natural numbers, so be careful there. This notation is usually called set builder notation. It tells use how to build a set by telling us precisely the condition elements must meet to gain access (the condition is the logical statement after the “:” symbol). Reading and comprehending sets written in this way takes practice. Here are some more examples: Example 0.3.1 Describe each of the following sets both in words and by listing out enough elements to see the pattern. 1. {x : x + 3 ∈ N}. 2. {x ∈ N : x + 3 ∈ N}. 3. {x : x ∈ N ∨ −x ∈ N}. 26 0. Introduction and Preliminaries 4. {x : x ∈ N ∧ −x ∈ N}. Solution. 1. This is the set of all numbers which are 3 less than a natural number (i.e., that if you add 3 to them, you get a natural number). The set could also be written as {−3, −2, −1, 0, 1, 2, . . .} (note that 0 is a natural number, so −3 is in this set because −3 + 3 0). 2. This is the set of all natural numbers which are 3 less than a natural number. So here we just have {0, 1, 2, 3 . . .}. 3. This is the set of all integers (positive and negative whole numbers, written Z). In other words, {. . . , −2, −1, 0, 1, 2, . . .}. 4. Here we want all numbers x such that x and −x are natural numbers. There is only one: 0. So we have the set {0}. There is also a subtle variation on set builder notation. While the condition is generally given after the “such that”, sometimes it is hidden in the ﬁrst part. Here is an example. Example 0.3.2 List a few elements in the sets below and describe them in words. The set Z is the set of integers; positive and negative whole numbers. 1. A {x ∈ Z : x 2 ∈ N} 2. B {x 2 : x ∈ N} Solution. 1. The set of integers that pass the condition that their square is a natural number. Well, every integer, when you square it, gives you a non-negative integer, so a natural number. Thus A Z {. . . , −2, −1, 0, 1, 2, 3, . . .}. 2. Here we are looking for the set of all x 2 s where x is a natural number. So this set is simply the set of perfect squares. B {0, 1, 4, 9, 16, . . .}. Another way we could have written this set, using more strict set builder notation, would be as B {x ∈ N : x n 2 for some n ∈ N}. 0.3. Sets 27 We already have a lot of notation, and there is more yet. Below is a handy chart of symbols. Some of these will be discussed in greater detail as we move forward. Special sets. ∅ The empty set is the set which contains no elements. U The universe set is the set of all elements. N The set of natural numbers. That is, N {0, 1, 2, 3 . . .}. Z The set of integers. That is, Z {. . . , −2, −1, 0, 1, 2, 3, . . .}. Q The set of rational numbers. R The set of real numbers. P(A) The power set of any set A is the set of all subsets of A. Set Theory Notation. {, } We use these braces to enclose the elements of a set. So {1, 2, 3} is the set containing 1, 2, and 3. : {x : x > 2} is the set of all x such that x is greater than 2. ∈ 2 ∈ {1, 2, 3} asserts that 2 is an element of the set {1, 2, 3}. < 4 < {1, 2, 3} because 4 is not an element of the set {1, 2, 3}. ⊆ A ⊆ B asserts that A is a subset of B: every element of A is also an element of B. ⊂ A ⊂ B asserts that A is a proper subset of B: every element of A is also an element of B, but A , B. ∩ A ∩ B is the intersection of A and B: the set containing all elements which are elements of both A and B. ∪ A ∪ B is the union of A and B: is the set containing all elements which are elements of A or B or both. × A × B is the Cartesian product of A and B: the set of all ordered pairs (a, b) with a ∈ A and b ∈ B. \ A \ B is A set-minus B: the set containing all elements of A which are not elements of B. A The complement of A is the set of everything which is not an element of A. |A| The cardinality (or size) of A is the number of elements in A. 28 0. Introduction and Preliminaries Investigate! 1. Find the cardinality of each set below. (a) A {3, 4, . . . , 15}. (b) B {n ∈ N : 2 < n ≤ 200}. (c) C {n ≤ 100 : n ∈ N ∧ ∃m ∈ N(n 2m + 1)}. 2. Find two sets A and B for which |A| 5, |B| 6, and |A ∪ B| 9. What is |A ∩ B|? 3. Find sets A and B with |A| |B| such that |A ∪ B| 7 and |A ∩ B| 3. What is |A|? 4. Let A {1, 2, . . . , 10}. Deﬁne B2 {B ⊆ A : |B| 2}. Find |B2 |. 5. For any sets A and B, deﬁne AB {ab : a ∈ A ∧ b ∈ B}. If A {1, 2} and B {2, 3, 4}, what is |AB|? What is |A × B|? ! Attempt the above activity before proceeding ! Relationships Between Sets We have already said what it means for two sets to be equal: they have exactly the same elements. Thus, for example, {1, 2, 3} {2, 1, 3}. (Remember, the order the elements are written down in does not matter.) Also, {1, 2, 3} {1, 1 + 1, 1 + 1 + 1} {I, II, III} {1, 2, 3, 1 + 2} since these are all ways to write the set containing the ﬁrst three positive integers (how we write them doesn’t matter, just what they are). What about the sets A {1, 2, 3} and B {1, 2, 3, 4}? Clearly A , B, but notice that every element of A is also an element of B. Because of this we say that A is a subset of B, or in symbols A ⊂ B or A ⊆ B. Both symbols are read “is a subset of.” The diﬀerence is that sometimes we want to say that A is either equal to or is a subset of B, in which case we use ⊆. This is analogous to the diﬀerence between < and ≤. 0.3. Sets 29 Example 0.3.3 Let A {1, 2, 3, 4, 5, 6}, B {2, 4, 6}, C {1, 2, 3} and D {7, 8, 9}. Determine which of the following are true, false, or meaningless. 1. A ⊂ B. 4. ∅ ∈ A. 7. 3 ∈ C. 2. B ⊂ A. 5. ∅ ⊂ A. 8. 3 ⊂ C. 3. B ∈ C. 6. A < D. 9. {3} ⊂ C. Solution. 1. False. For example, 1 ∈ A but 1 < B. 2. True. Every element in B is an element in A. 3. False. The elements in C are 1, 2, and 3. The set B is not equal to 1, 2, or 3. 4. False. A has exactly 6 elements, and none of them are the empty set. 5. True. Everything in the empty set (nothing) is also an element of A. Notice that the empty set is a subset of every set. 6. Meaningless. A set cannot be less than another set. 7. True. 3 is one of the elements of the set C. 8. Meaningless. 3 is not a set, so it cannot be a subset of another set. 9. True. 3 is the only element of the set {3}, and is an element of C, so every element in {3} is an element of C. In the example above, B is a subset of A. You might wonder what other sets are subsets of A. If you collect all these subsets of A into a new set, we get a set of sets. We call the set of all subsets of A the power set of A, and write it P(A). Example 0.3.4 Let A {1, 2, 3}. Find P(A). Solution. P(A) is a set of sets, all of which are subsets of A. So P(A) {∅, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}}. 30 0. Introduction and Preliminaries Notice that while 2 ∈ A, it is wrong to write 2 ∈ P(A) since none of the elements in P(A) are numbers! On the other hand, we do have {2} ∈ P(A) because {2} ⊆ A. What does a subset of P(A) look like? Notice that {2} ⊈ P(A) because not everything in {2} is in P(A). But we do have {{2}} ⊆ P(A). The only element of {{2}} is the set {2} which is also an element of P(A). We could take the collection of all subsets of P(A) and call that P(P(A)). Or even the power set of that set of sets of sets. Another way to compare sets is by their size. Notice that in the example above, A has 6 elements and B, C, and D all have 3 elements. The size of a set is called the set’s cardinality . We would write |A| 6, |B| 3, and so on. For sets that have a ﬁnite number of elements, the cardinality of the set is simply the number of elements in the set. Note that the cardinality of {1, 2, 3, 2, 1} is 3. We do not count repeats (in fact, {1, 2, 3, 2, 1} is exactly the same set as {1, 2, 3}). There are sets with inﬁnite cardinality, such as N, the set of rational numbers (written Q), the set of even natural numbers, and the set of real numbers (R). It is possible to distinguish between diﬀerent inﬁnite cardinalities, but that is beyond the scope of this text. For us, a set will either be inﬁnite, or ﬁnite; if it is ﬁnite, the we can determine its cardinality by counting elements. Example 0.3.5 1. Find the cardinality of A {23, 24, . . . , 37, 38}. 2. Find the cardinality of B {1, {2, 3, 4}, ∅}. 3. If C {1, 2, 3}, what is the cardinality of P(C)? Solution. 1. Since 38 − 23 15, we can conclude that the cardinality of the set is |A| 16 (you need to add one since 23 is included). 2. Here |B| 3. The three elements are the number 1, the set {2, 3, 4}, and the empty set. 3. We wrote out the elements of the power set P(C) above, and there are 8 elements (each of which is a set). So |P(C)| 8. (You might wonder if there is a relationship between |A| and |P(A)| for all sets A. This is a good question which we will return to in Chapter 1.) 0.3. Sets 31 Operations On Sets Is it possible to add two sets? Not really, however there is something similar. If we want to combine two sets to get the collection of objects that are in either set, then we can take the union of the two sets. Symbolically, C A ∪ B, read, “C is the union of A and B,” means that the elements of C are exactly the elements which are either an element of A or an element of B (or an element of both). For example, if A {1, 2, 3} and B {2, 3, 4}, then A ∪ B {1, 2, 3, 4}. The other common operation on sets is intersection. We write, C A∩B and say, “C is the intersection of A and B,” when the elements in C are precisely those both in A and in B. So if A {1, 2, 3} and B {2, 3, 4}, then A ∩ B {2, 3}. Often when dealing with sets, we will have some understanding as to what “everything” is. Perhaps we are only concerned with natural numbers. In this case we would say that our universe is N. Sometimes we denote this universe by U. Given this context, we might wish to speak of all the elements which are not in a particular set. We say B is the complement of A, and write, BA when B contains every element not contained in A. So, if our universe is {1, 2, . . . , 9, 10}, and A {2, 3, 5, 7}, then A {1, 4, 6, 8, 9, 10}. Of course we can perform more than one operation at a time. For example, consider A ∩ B. This is the set of all elements which are both elements of A and not elements of B. What have we done? We’ve started with A and removed all of the elements which were in B. Another way to write this is the set diﬀerence: A ∩ B A \ B. It is important to remember that these operations (union, intersection, complement, and diﬀerence) on sets produce other sets. Don’t confuse these with the symbols from the previous section (element of and subset of). A ∩ B is a set, while A ⊆ B is true or false. This is the same diﬀerence as between 3 + 2 (which is a number) and 3 ≤ 2 (which is false). 32 0. Introduction and Preliminaries Example 0.3.6 Let A {1, 2, 3, 4, 5, 6}, B {2, 4, 6}, C {1, 2, 3} and D {7, 8, 9}. If the universe is U {1, 2, . . . , 10}, ﬁnd: 1. A ∪ B. 4. A ∩ D. 7. (D∩C)∪A ∩ B. 2. A ∩ B. 5. B ∪ C. 8. ∅ ∪ C. 3. B ∩ C. 6. A \ B. 9. ∅ ∩ C. Solution. 1. A ∪ B {1, 2, 3, 4, 5, 6} A since everything in B is already in A. 2. A ∩ B {2, 4, 6} B since everything in B is in A. 3. B ∩ C {2} as the only element of both B and C is 2. 4. A ∩ D ∅ since A and D have no common elements. 5. B ∪ C {5, 7, 8, 9, 10}. First we ﬁnd that B∪C {1, 2, 3, 4, 6}, then we take everything not in that set. 6. A \ B {1, 3, 5} since the elements 1, 3, and 5 are in A but not in B. This is the same as A ∩ B. 7. (D ∩ C) ∪ A ∩ B {1, 3, 5, 7, 8, 9, 10}. The set contains all elements that are either in D but not in C (i.e., {7, 8, 9}), or not in both A and B (i.e., {1, 3, 5, 7, 8, 9, 10}). 8. ∅ ∪ C C since nothing is added by the empty set. 9. ∅ ∩ C ∅ since nothing can be both in a set and in the empty set. Having notation like this is useful. We will often want to add or remove elements from sets, and our notation allows us to do so precisely. Example 0.3.7 If A {1, 2, 3}, then we can describe the set we get by adding the number 4 as A ∪ {4}. If we want to express the set we get by removing the number 2 from A we can do so by writing A \ {2}. Careful though. If you add an element to the set, you get a new set! So you would have B A ∪ {4} and then correctly say that B contains 4, but A does not. 0.3. Sets 33 You might notice that the symbols for union and intersection slightly resemble the logic symbols for “or” and “and.” This is no accident. What does it mean for x to be an element of A ∪ B? It means that x is an element of A or x is an element of B (or both). That is, x ∈A∪B ⇔ x ∈ A ∨ x ∈ B. x ∈A∩B ⇔ x ∈ A ∧ x ∈ B. x∈A ⇔ ¬(x ∈ A). Similarly, Also, which says x is an element of the complement of A if x is not an element of A. There is one more way to combine sets which will be useful for us: the Cartesian product, A × B. This sounds fancy but is nothing you haven’t seen before. When you graph a function in calculus, you graph it in the Cartesian plane. This is the set of all ordered pairs of real numbers (x, y). We can do this for any pair of sets, not just the real numbers with themselves. Put another way, A × B {(a, b) : a ∈ A ∧ b ∈ B}. The ﬁrst coordinate comes from the ﬁrst set and the second coordinate comes from the second set. Sometimes we will want to take the Cartesian product of a set with itself, and this is ﬁne: A × A {(a, b) : a, b ∈ A} (we might also write A2 for this set). Notice that in A × A, we still want all ordered pairs, not just the ones where the ﬁrst and second coordinate are the same. We can also take products of 3 or more sets, getting ordered triples, or quadruples, and so on. Example 0.3.8 Let A {1, 2} and B {3, 4, 5}. Find A × B and A × A. How many elements do you expect to be in B × B? Solution. A × B {(1, 3), (1, 4), (1, 5), (2, 3), (2, 4), (2, 5)}. A × A A2 {(1, 1), (1, 2), (2, 1), (2, 2)}. |B × B| 9. There will be 3 pairs with ﬁrst coordinate 3, three more with ﬁrst coordinate 4, and a ﬁnal three with ﬁrst coordinate 5. Venn Diagrams There is a very nice visual tool we can use to represent operations on sets. A Venn diagram displays sets as intersecting circles. We can shade the region we are talking about when we carry out an operation. We can 34 0. Introduction and Preliminaries also represent cardinality of a particular set by putting the number in the corresponding region. A A B B C Each circle represents a set. The rectangle containing the circles represents the universe. To represent combinations of these sets, we shade the corresponding region. For example, we could draw A ∩ B as: A B Here is a representation of A ∩ B, or equivalently A \ B: A B A more complicated example is (B ∩ C) ∪ (C ∩ A), as seen below. A B C Notice that the shaded regions above could also be arrived at in another way. We could have started with all of C, then excluded the region where C and A overlap outside of B. That region is (A ∩ C) ∩ B. So the above Venn diagram also represents C ∩ (A ∩ C) ∩ B . So using just the picture, we have determined that (B ∩ C) ∪ (C ∩ A) C ∩ (A ∩ C) ∩ B . 0.3. Sets 35 Exercises 1. Let A {1, 4, 9} and B {1, 3, 6, 10}. Find each of the following sets. (a) A ∪ B. (b) A ∩ B. (c) A \ B. (d) B \ A. 2. Find the least element of each of the following sets, if there is one. (a) {n ∈ N : n 2 − 3 ≥ 2}. (b) {n ∈ N : n 2 − 5 ∈ N}. (c) {n 2 + 1 : n ∈ N}. (d) {n ∈ N : n k 2 + 1 for some k ∈ N}. 3. Find the following cardinalities: (a) |A| when A {4, 5, 6, . . . , 37}. (b) |A| when A {x ∈ Z : −2 ≤ x ≤ 100}. (c) |A ∩ B| when A {x ∈ N : x ≤ 20} and B {x ∈ N : x is prime}. 4. Find a set of largest possible size that is a subset of both {1, 2, 3, 4, 5} and {2, 4, 6, 8, 10}. 5. Find a set of smallest possible size that has both {1, 2, 3, 4, 5} and {2, 4, 6, 8, 10} as subsets. 6. Let A {n ∈ N : 20 ≤ n < 50} and B {n ∈ N : 10 < n ≤ 30}. Suppose C is a set such that C ⊆ A and C ⊆ B. What is the largest possible cardinality of C? 7. Let A {1, 2, 3, 4, 5} and B {2, 3, 4}. How many sets C have the property that C ⊆ A and B ⊆ C. 8. Let A {1, 2, 3, 4, 5}, B {3, 4, 5, 6, 7}, and C {2, 3, 5}. (a) Find A ∩ B. (b) Find A ∪ B. (c) Find A \ B. (d) Find A ∩ (B ∪ C). 9. Let A {x ∈ N : 4 ≤ x < 12} and B {x ∈ N : x is even}. (a) Find A ∩ B. (b) Find A \ B. 36 0. Introduction and Preliminaries 10. Let A {x ∈ N : 3 ≤ x ≤ 13}, B {x ∈ N : x is even}, and C {x ∈ N : x is odd}. (a) Find A ∩ B. (b) Find A ∪ B. (c) Find B ∩ C. (d) Find B ∪ C. 11. Find an example of sets A and B such that A ∩ B {3, 5} and A ∪ B {2, 3, 5, 7, 8}. 12. Find an example of sets A and B such that A ⊆ B and A ∈ B. 13. Recall Z {. . . , −2, −1, 0, 1, 2, . . .} (the integers). Let Z+ {1, 2, 3, . . .} be the positive integers. Let 2Z be the even integers, 3Z be the multiples of 3, and so on. (a) Is Z+ ⊆ 2Z? Explain. (b) Is 2Z ⊆ Z+ ? Explain. (c) Find 2Z ∩ 3Z. Describe the set in words, and using set notation. (d) Express {x ∈ Z : ∃y ∈ Z(x 2y ∨ x 3y)} as a union or intersection of two sets already described in this problem. 14. Let A2 be the set of all multiples of 2 except for 2. Let A3 be the set of all multiples of 3 except for 3. And so on, so that A n is the set of all multiples of n except for n, for any n ≥ 2. Describe (in words) the set A2 ∪ A3 ∪ A4 ∪ · · ·. 15. Draw a Venn diagram to represent each of the following: (a) A ∪ B (b) (A ∪ B) (c) A ∩ (B ∪ C) (d) (A ∩ B) ∪ C (e) A ∩ B ∩ C (f) (A ∪ B) \ C 16. Describe a set in terms of A and B (using set notation) which has the following Venn diagram: A B 0.3. Sets 37 17. Let A {a, b, c, d}. Find P(A). 18. Let A {1, 2, . . . , 10}. How many subsets of A contain exactly one element (i.e., how many singleton subsets are there)? How many doubleton subsets (containing exactly two elements) are there? 19. Let A {1, 2, 3, 4, 5, 6}. Find all sets B ∈ P(A) which have the property {2, 3, 5} ⊆ B. 20. Find an example of sets A and B such that |A| 4, |B| 5, and |A ∪ B| 9. 21. Find an example of sets A and B such that |A| 3, |B| 4, and |A ∪ B| 5. 22. Are there sets A and B such that |A| |B|, |A∪B| 10, and |A∩B| 5? Explain. 23. Let A {2, 4, 6, 8}. Suppose B is a set with |B| 5. (a) What are the smallest and largest possible values of |A ∪ B|? Explain. (b) What are the smallest and largest possible values of |A ∩ B|? Explain. (c) What are the smallest and largest possible values of |A × B|? Explain. 24. Let X {n ∈ N : 10 ≤ n < 20}. Find examples of sets with the properties below and very brieﬂy explain why your examples work. (a) A set A ⊆ N with |A| 10 such that X \ A {10, 12, 14}. (b) A set B ∈ P(X) with |B| 5. (c) A set C ⊆ P(X) with |C| 5. (d) A set D ⊆ X × X with |D| 5 (e) A set E ⊆ X such that |E| ∈ E. 25. Let A, B and C be sets. (a) Suppose that A ⊆ B and B ⊆ C. Does this mean that A ⊆ C? Prove your answer. Hint: to prove that A ⊆ C you must prove the implication, “for all x, if x ∈ A then x ∈ C.” (b) Suppose that A ∈ B and B ∈ C. Does this mean that A ∈ C? Give an example to prove that this does NOT always happen (and explain why your example works). You should be able to give an example where |A| |B| |C| 2. 38 0. Introduction and Preliminaries 26. In a regular deck of playing cards there are 26 red cards and 12 face cards. Explain, using sets and what you have learned about cardinalities, why there are only 32 cards which are either red or a face card. 27. Find an example of a set A with |A| 3 which contains only other sets and has the following property: for all sets B ∈ A, we also have B ⊆ A. Explain why your example works. (FYI: sets that have this property are called transitive.) 28. Consider the sets A and B, where A {3, |B|} and B {1, |A|, |B|}. What are the sets? 29. Explain why there is no set A which satisﬁes A {2, |A|}. 30. Find all sets A, B, and C which satisfy the following. A {1, |B| , |C|} B {2, |A| , |C|} C {1, 2, |A| , |B|}. 0.4. Functions 0.4 39 Functions A function is a rule that assigns each input exactly one output. We call the output the image of the input. The set of all inputs for a function is called the domain. The set of all allowable outputs is called the codomain. We would write f : X → Y to describe a function with name f , domain X and codomain Y. This does not tell us which function f is though. To deﬁne the function, we must describe the rule. This is often done by giving a formula to compute the output for any input (although this is certainly not the only way to describe the rule). For example, consider the function f : N → N deﬁned by f (x) x 2 +3. Here the domain and codomain are the same set (the natural numbers). The rule is: take your input, multiply it by itself and add 3. This works because we can apply this rule to every natural number (every element of the domain) and the result is always a natural number (an element of the codomain). Notice though that not every natural number is actually an output (there is no way to get 0, 1, 2, 5, etc.). The set of natural numbers that are outputs is called the range of the function (in this case, the range is {3, 4, 7, 12, 19, 28, . . .}, all the natural numbers that are 3 more than a perfect square). The key thing that makes a rule a function is that there is exactly one output for each input. That is, it is important that the rule be a good rule. What output do we assign to the input 7? There can only be one answer for any particular function. Example 0.4.1 The following are all examples of functions: 1. f : Z → Z deﬁned by f (n) 3n. The domain and codomain are both the set of integers. However, the range is only the set of integer multiples of 3. 2. g : {1, 2, 3} → {a, b, c} deﬁned by g(1) c, g(2) a and g(3) a. The domain is the set {1, 2, 3}, the codomain is the set {a, b, c} and the range is the set {a, c}. Note that g(2) and g(3) are the same element of the codomain. This is okay since each element in the domain still has only one output. 3. h : {1, 2, 3, 4} → N deﬁned by the table: x h(x) 1 3 2 6 3 9 4 12 Here the domain is the ﬁnite set {1, 2, 3, 4} and to codomain is the set of natural numbers, N. At ﬁrst you might think this 40 0. Introduction and Preliminaries function is the same as f deﬁned above. It is absolutely not. Even though the rule is the same, the domain and codomain are diﬀerent, so these are two diﬀerent functions. Example 0.4.2 Just because you can describe a rule in the same way you would write a function, does not mean that the rule is a function. The following are NOT functions. 1. f : N → N deﬁned by f (n) n2 . The reason this is not a function is because not every input has an output. Where does f send 3? The rule says that f (3) 32 , but 32 is not an element of the codomain. 2. Consider the rule that matches each person to their phone number. If you think of the set of people as the domain and the set of phone numbers as the codomain, then this is not a function, since some people have two phone numbers. Switching the domain and codomain sets doesn’t help either, since some phone numbers belong to multiple people (assuming some households still have landlines when you are reading this). Describing Functions It is worth making a distinction between a function and its description. The function is the abstract mathematical object that in some way exists whether or not anyone ever talks about it. But when we do want to talk about the function, we need a way to describe it. A particular function can be described in multiple ways. Some calculus textbooks talk about the Rule of Four, that every function can be described in four ways: algebraically (a formula), numerically (a table), graphically, or in words. In discrete math, we can still use any of these to describe functions, but we can also be more speciﬁc since we are primarily concerned with functions that have N or a ﬁnite subset of N as their domain. Describing a function graphically usually means drawing the graph of the function: plotting the points on the plane. We can do this, and might get a graph like the following for a function f : {1, 2, 3} → {1, 2, 3}. 0.4. Functions 41 It would be absolutely WRONG to connect the dots or try to ﬁt them to some curve. There are only three elements in the domain. A curve would mean that the domain contains an entire interval of real numbers. Here is another way to represent that same function: 1 2 3 1 2 3 This shows that the function f sends 1 to 2, 2 to 1 and 3 to 3: just follow the arrows. The arrow diagram used to deﬁne the function above can be very helpful in visualizing functions. We will often be working with functions with ﬁnite domains, so this kind of picture is often more useful than a traditional graph of a function. Note that for ﬁnite domains, ﬁnding an algebraic formula that gives the output for any input is often impossible. Of course we could use a piecewise deﬁned function, like x+1 if x 1 x if x 3 f (x) x − 1 if x 2 . This describes exactly the same function as above, but we can all agree is a ridiculous way of doing so. Since we will so often use functions with small domains and codomains, let’s adopt some notation to describe them. All we need is some clear way of denoting the image of each element in the domain. In fact, writing a table of values would work perfectly: x f (x) 0 3 1 3 2 2 3 4 4 1 We simplify this further by writing this as a matrix with each input directly over its output: 0 1 2 3 4 f . 3 3 2 4 1 42 0. Introduction and Preliminaries Note this is just notation and not the same sort of matrix you would ﬁnd in a linear algebra class (it does not make sense to do operations with these matrices, or row reduce them, for example). One advantage of the two-line notation over the arrow diagrams is that it is harder to accidentally deﬁne a rule that is not a function using two-line notation. Example 0.4.3 Which of the following diagrams represent a function? Let X {1, 2, 3, 4} and Y {a, b, c, d}. 1 a f :X→Y 3 2 b c 4 1 d a g:X→Y 3 2 b c 4 1 d a h:X→Y 3 2 b c 4 d Solution. f is a function. So is g. There is no problem with an element of the codomain not being the image of any input, and there is no problem with a from the codomain being the image of both 2 and 3 from the domain. We could use our two-line notation to write these as 1 2 3 4 f d a c b 1 2 3 4 g . d a a b However, h is NOT a function. In fact, it fails for two reasons. First, the element 1 from the domain has not been mapped to any element from the codomain. Second, the element 2 from the domain has been mapped to more than one element from the codomain (a and c). Note that either one of these problems is enough to make a rule not a function. In general, neither of the following mappings are functions: It might also be helpful to think about how you would write the two-line notation for h. We would have something like: h 1 2 3 4 . a, c? d b There is nothing under 1 (bad) and we needed to put more than one thing under 2 (very bad). With a rule that is actually a function, the two-line notation will always “work”. 0.4. Functions 43 We will also be interested in functions with domain N. Here twoline notation is no good, but describing the function algebraically is often possible. Even tables are a little awkward, since they do not describe the function completely. For example, consider the function f : N → N given by the table below. x f (x) 0 0 1 1 2 4 3 9 4 16 5 25 ... ... Have I given you enough entries for you to be able to determine f (6)? You might guess that f (6) 36, but there is no way for you to know this for sure. Maybe I am being a jerk and intended f (6) 42. In fact, for every natural number n, there is a function that agrees with the table above, but for which f (6) n. Okay, suppose I really did mean for f (6) 36, and in fact, for the rule that you think is governing the function to actually be the rule. Then I should say what that rule is. f (n) n 2 . Now there is no confusion possible. Giving an explicit formula that calculates the image of any element in the domain is a great way to describe a function. We will say that these explicit rules are closed formulas for the function. There is another very useful way to describe functions whose domain is N, that rely speciﬁcally on the structure of the natural numbers. We can deﬁne a function recursively! Example 0.4.4 Consider the function f : N → N given by f (0) 0 and f (n + 1) f (n) + 2n + 1. Find f (6). Solution. The rule says that f (6) f (5) + 11 (we are using 6 n + 1 so n 5). We don’t know what f (5) is though. Well, we know that f (5) f (4) + 9. So we need to compute f (4), which will require knowing f (3), which will require f (2),. . . will it ever end? Yes! In fact, this process will always end because we have N as our domain, so there is a least element. And we gave the value of f (0) explicitly, so we are good. In fact, we might decide to work up to f (6) instead of working down from f (6): f (1) f (0) + 1 0+11 f (2) f (1) + 3 1+34 f (3) f (2) + 5 4+59 f (4) f (3) + 7 9 + 7 16 f (5) f (4) + 9 16 + 9 25 44 0. Introduction and Preliminaries f (6) f (5) + 11 25 + 11 36 It looks that this recursively deﬁned function is the same as the explicitly deﬁned function f (n) n 2 . Is it? Later we will prove that it is. Recursively deﬁned functions are often easier to create from a “real world” problem, because they describe how the values of the functions are changing. However, this comes with a price. It is harder to calculate the image of a single input, since you need to know the images of other (previous) elements in the domain. Recursively Deﬁned Functions. For a function f : N → N, a recursive deﬁnition consists of an initial condition together with a recurrence relation. The initial condition is the explicitly given value of f (0). The recurrence relation is a formula for f (n + 1) in terms for f (n) (and possibly n itself). Example 0.4.5 Give recursive deﬁnitions for the functions described below. 1. f : N → N gives the number of snails in your terrarium n years after you built it, assuming you started with 3 snails and the number of snails doubles each year. 2. g : N → N gives the number of push-ups you do n days after you started your push-ups challenge, assuming you could do 7 push-ups on day 0 and you can do 2 more push-ups each day. 3. h : N → N deﬁned by f (n) n!. Recall that n! 1 · 2 · 3 · · · · · (n − 1) · n is the product of all numbers from 1 through n. We also deﬁne 0! 1. Solution. 1. The initial condition is f (0) 3. To get f (n + 1) we would double the number of snails in the terrarium the previous year, which is given by f (n). Thus f (n + 1) 2 f (n). The full recursive deﬁnition contains both of these, and would be written, f (0) 3; f (n + 1) 2 f (n). 0.4. Functions 45 2. We are told that on day 0 you can do 7 push-ups, so f (0) 7. The number of push-ups you can do on day n + 1 is 2 more than the number you can do on day n, which is given by f (n). Thus f (0) 7; f (n + 1) f (n) + 2. 3. Here f (0) 1. To get the recurrence relation, think about how you can get f (n + 1) (n + 1)! from f (n) n!. If you write out both of these as products, you see that (n + 1)! is just like n! except you have one more term in the product, an extra n + 1. So we have, f (0) 1; f (n + 1) (n + 1) · f (n). Surjections, Injections, and Bijections We now turn to investigating special properties functions might or might not possess. In the examples above, you may have noticed that sometimes there are elements of the codomain which are not in the range. When this sort of the thing does not happen, (that is, when everything in the codomain is in the range) we say the function is onto or that the function maps the domain onto the codomain. This terminology should make sense: the function puts the domain (entirely) on top of the codomain. The fancy math term for an onto function is a surjection, and we say that an onto function is a surjective function. In pictures: Surjective Not surjective Example 0.4.6 Which functions are surjective (i.e., onto)? 1. f : Z → Z deﬁned by f (n) 3n. 1 2 3 2. g : {1, 2, 3} → {a, b, c} deﬁned by g . c a a 3. h : {1, 2, 3} → {1, 2, 3} deﬁned as follows: 46 0. Introduction and Preliminaries 1 2 3 1 2 3 Solution. 1. f is not surjective. There are elements in the codomain which are not in the range. For example, no n ∈ Z gets mapped to the number 1 (the rule would say that 13 would be sent to 1, but 31 is not in the domain). In fact, the range of the function is 3Z (the integer multiples of 3), which is not equal to Z. 2. g is not surjective. There is no x ∈ {1, 2, 3} (the domain) for which g(x) b, so b, which is in the codomain, is not in the range. Notice that there is an element from the codomain “missing” from the bottom row of the matrix. 3. h is surjective. Every element of the codomain is also in the range. Nothing in the codomain is missed. To be a function, a rule cannot assign a single element of the domain to two or more diﬀerent elements of the codomain. However, we have seen that the reverse is permissible: a function might assign the same element of the codomain to two or more diﬀerent elements of the domain. When this does not occur (that is, when each element of the codomain is the image of at most one element of the domain) then we say the function is one-to-one. Again, this terminology makes sense: we are sending at most one element from the domain to one element from the codomain. One input to one output. The fancy math term for a one-to-one function is an injection. We call one-to-one functions injective functions. In pictures: Injective Not injective Example 0.4.7 Which functions are injective (i.e., one-to-one)? 1. f : Z → Z deﬁned by f (n) 3n. 0.4. Functions 47 1 2 3 2. g : {1, 2, 3} → {a, b, c} deﬁned by g . c a a 3. h : {1, 2, 3} → {1, 2, 3} deﬁned as follows: 1 2 3 1 2 3 Solution. 1. f is injective. Each element in the codomain is assigned to at most one element from the domain. If x is a multiple of three, then only x/3 is mapped to x. If x is not a multiple of 3, then there is no input corresponding to the output x. 2. g is not injective. Both inputs 2 and 3 are assigned the output a. Notice that there is an element from the codomain that appears more than once on the bottom row of the matrix. 3. h is injective. Each output is only an output once. Be careful: “surjective” and “injective” are NOT opposites. You can see in the two examples above that there are functions which are surjective but not injective, injective but not surjective, both, or neither. In the case when a function is both one-to-one and onto (an injection and surjection), we say the function is a bĳection, or that the function is a bĳective function. To illustrate the contrast between these two properties, consider a more formal deﬁnition of each, side by side. Injective vs Surjective. A function is injective provided every element of the codomain is the image of at most one element from the domain. A function is surjective provided every element of the codomain is the image of at least one element from the domain. Notice both properties are determined by what happens to elements of the codomain: they could be repeated as images or they could be “missed” (not be images). Injective functions do not have repeats but might or might not miss elements. Surjective functions do not miss elements, but might 48 0. Introduction and Preliminaries or might not have repeats. The bĳective functions are those that do not have repeats and do not miss elements. Image and Inverse Image When discussing functions, we have notation for talking about an element of the domain (say x) and its corresponding element in the codomain (we write f (x), which is the image of x). Sometimes we will want to talk about all the elements that are images of some subset of the domain. It would also be nice to start with some element of the codomain (say y) and talk about which element or elements (if any) from the domain it is the image of. We could write “those x in the domain such that f (x) y,” but this is a lot of writing. Here is some notation to make our lives easier. To address the ﬁrst situation, what we are after is a way to describe the set of images of elements in some subset of the domain. Suppose f : X → Y is a function and that A ⊆ X is some subset of the domain (possibly all of it). We will use the notation f (A) to denote the image of A under f , namely the set of elements in Y that are the image of elements from A. That is, f (A) { f (a) ∈ Y : a ∈ A}. We can do this in the other direction as well. We might ask which elements of the domain get mapped to a particular set in the codomain. Let f : X → Y be a function and suppose B ⊆ Y is a subset of the codomain. Then we will write f −1 (B) for the inverse image of B under f , namely the set of elements in X whose image are elements in B. In other words, f −1 (B) {x ∈ X : f (x) ∈ B}. Often we are interested in the element(s) whose image is a particular element y of in the codomain. The notation above works: f −1 ({ y}) is the set of all elements in the domain that f sends to y. It makes sense to think of this as a set: there might not be anything sent to y (if y is not in the range), in which case f −1 ({ y}) ∅. Or f might send multiple elements to y (if f is not injective). As a notational convenience, we usually drop the set braces around the y and write f inf(y) instead for this set. WARNING: f −1 (y) is not an inverse function! Inverse functions only exist for bĳections, but f −1 (y) is deﬁned for any function f . The point: f −1 (y) is a set, not an element of the domain. This is just sloppy notation for f −1 ({ y}). To help make this distinction, we would call f −1 (y) the complete inverse image of y under f . It is not the image of y under f −1 (since the function f −1 might not exist). 0.4. Functions 49 Example 0.4.8 Consider the function f : {1, 2, 3, 4, 5, 6} → {a, b, c, d} given by 1 2 3 4 5 6 f . a a b b b c Find f ({1, 2, 3}), f inf({a, b}), and f −1 (d). Solution. f ({1, 2, 3}) {a, b} since a and b are the elements in the codomain to which f sends 1 and 2. f inf({a, b}) {1, 2, 3, 4, 5} since these are exactly the elements that f sends to a and b. f inf(d) ∅ since d is not in the range of f . Example 0.4.9 Consider the function g : Z → Z deﬁned by g(n) n 2 + 1. Find g(1) and g({1}). Then ﬁnd g −1 (1), g −1 (2), and g −1 (3). Solution. Note that g(1) , g({1}). The ﬁrst is an element: g(1) 2. The second is a set: g({1}) {2}. To ﬁnd g −1 (1), we need to ﬁnd all integers n such that n 2 +1 1. Clearly only 0 works, so g −1 (1) {0} (note that even though there is only one element, we still write it as a set with one element in it). To ﬁnd g −1 (2), we need to ﬁnd all n such that n 2 + 1 2. We see g −1 (2) {−1, 1}. Finally, if n 2 + 1 3, then we are looking for an n such that n 2 2. There are no such integers so g −1 (3) ∅. Since f −1 (y) is a set, it makes sense to ask for f −1 (y) , the number of elements in the domain which map to y. Example 0.4.10 Find a function f : {1, 2, 3, 4, 5} → N such that f −1 (7) 5. Solution. There is only one such function. We need ﬁve elements of the domain to map to the number 7 ∈ N. Since there are only ﬁve elements in the domain, all of them must map to 7. So 1 2 3 4 5 . f 7 7 7 7 7 50 0. Introduction and Preliminaries Function Definitions. Here is a summary of all the main concepts and deﬁnitions we use when working with functions. • A function is a rule that assigns each element of a set, called the domain, to exactly one element of a second set, called the codomain. • Notation: f : X → Y is our way of saying that the function is called f , the domain is the set X, and the codomain is the set Y. • To specify the rule for a function with small domain, use two-line notation by writing a matrix with each output directly below its corresponding input, as in: 1 2 3 4 f . 2 1 3 1 • f (x) y means the element x of the domain (input) is assigned to the element y of the codomain. We say y is an output. Alternatively, we call y the image of x under f . • The range is a subset of the codomain. It is the set of all elements which are assigned to at least one element of the domain by the function. That is, the range is the set of all outputs. • A function is injective (an injection or one-to-one) if every element of the codomain is the image of at most one element from the domain. • A function is surjective (a surjection or onto) if every element of the codomain is the image of at least one element from the domain. • A bĳection is a function which is both an injection and surjection. In other words, if every element of the codomain is the image of exactly one element from the domain. • The image of an element x in the domain is the element y in the codomain that x is mapped to. That is, the image of x under f is f (x). • The complete inverse image of an element y in the codomain, written f −1 (y), is the set of all elements in the domain which are assigned to y by the function. • The image of a subset A of the domain is the set f (A) { f (a) ∈ Y : a ∈ A}. • The inverse image of a a subset B of the codomain is the set f −1 (B) {x ∈ X : f (x) ∈ B}. 0.4. Functions 51 Exercises 1. Consider the function f : {1, 2, 3, 4} → {1, 2, 3, 4} given by 1 2 3 4 f (n) . 4 1 3 4 (a) Find f (1). (b) Find an element n in the domain such that f (n) 1. (c) Find an element n of the domain such that f (n) n. (d) Find an element of the codomain that is not in the range. 2. The following functions all have {1, 2, 3, 4, 5} as both their domain and codomain. For each, determine whether it is (only) injective, (only) surjective, bĳective, or neither injective nor surjective. 1 2 3 4 5 (a) f . 3 3 3 3 3 1 2 3 4 5 (b) f . 2 3 1 5 4 (c) f (x) 6 − x. ( (d) f (x) 3. x/2 if x is even (x + 1)/2 if x is odd . The following functions all have domain {1, 2, 3, 4, 5} and codomain {1, 2, 3}. For each, determine whether it is (only) injective, (only) surjective, bĳective, or neither injective nor surjective. 1 2 3 4 5 (a) f . 1 2 1 2 1 1 2 3 4 5 . (b) f 1 2 3 1 2 ( (c) f (x) 4. x if x ≤ 3 x−3 if x > 3 . The following functions all have domain {1, 2, 3, 4} and codomain {1, 2, 3, 4, 5}. For each, determine whether it is (only) injective, (only) surjective, bĳective, or neither injective nor surjective. 1 2 3 4 . (a) f 1 2 5 4 52 0. Introduction and Preliminaries 1 2 3 4 (b) f . 1 2 3 2 (c) f (x) gives the number of letters in the English word for the number x. For example, f (1) 3 since “one” contains three letters. 5. Write out all functions f : {1, 2, 3} → {a, b} (using two-line notation). How many functions are there? How many are injective? How many are surjective? How many are bĳective? 6. Write out all functions f : {1, 2} → {a, b, c} (in two-line notation). How many functions are there? How many are injective? How many are surjective? How many are bĳective? 7. Consider the function f : {1, 2, 3, 4, 5} → {1, 2, 3, 4} given by the table below: x 1 2 3 4 5 f (x) 3 2 4 1 2 (a) Is f injective? Explain. (b) Is f surjective? Explain. (c) Write the function using two-line notation. 8. Consider the function f : {1, 2, 3, 4} → {1, 2, 3, 4} given by the graph below. f (x) 4 3 2 1 1 2 3 4 x (a) Is f injective? Explain. (b) Is f surjective? Explain. (c) Write the function using two-line notation. 0.4. Functions 9. 53 Consider the function f : N → N given recursively by f (0) 1 and f (n + 1) 2 · f (n). Find f (10). 10. Suppose f : N → N satisﬁes the recurrence f (n + 1) f (n) + 3. Note that this is not enough information to deﬁne the function, since we don’t have an initial condition. For each of the initial conditions below, ﬁnd the value of f (5). (a) f (0) 0. (b) f (0) 1. (c) f (0) 2. (d) f (0) 100. 11. Suppose f : N → N satisﬁes the recurrence relation ( f (n) f (n + 1) 2 3 f (n) + 1 if f (n) is even if f (n) is odd . Note that with the initial condition f (0) 1, the values of the function are: f (1) 4, f (2) 2, f (3) 1, f (4) 4, and so on, the images cycling through those three numbers. Thus f is NOT injective (and also certainly not surjective). Might it be under other initial conditions?3 (a) If f satisﬁes the initial condition f (0) 5, is f injective? Explain why or give a speciﬁc example of two elements from the domain with the same image. (b) If f satisﬁes the initial condition f (0) 3, is f injective? Explain why or give a speciﬁc example of two elements from the domain with the same image. (c) If f satisﬁes the initial condition f (0) 27, then it turns out that f (105) 10 and no two numbers less than 105 have the same image. Could f be injective? Explain. (d) Prove that no matter what initial condition you choose, the function cannot be surjective. 12. For each function given below, determine whether or not the function is injective and whether or not the function is surjective. (a) f : N → N given by f (n) n + 4. (b) f : Z → Z given by f (n) n + 4. 3It turns out this is a really hard question to answer in general. The Collatz conjecture is that no matter what the initial condition is, the function will eventually produce 1 as an output. This is an open problem in mathematics: nobody knows the answer. 54 0. Introduction and Preliminaries (c) f : Z → Z given by f (n) 5n − 8. ( (d) f : Z → Z given by f (n) n/2 if n is even (n + 1)/2 if n is odd. 13. Let A {1, 2, 3, . . . , 10}. Consider the function f : P(A) → N given by f (B) |B|. That is, f takes a subset of A as an input and outputs the cardinality of that set. (a) Is f injective? Prove your answer. (b) Is f surjective? Prove your answer. (c) Find f −1 (1). (d) Find f −1 (0). (e) Find f −1 (12). 14. Let X {n ∈ N : 0 ≤ n ≤ 999} be the set of all numbers with three or fewer digits. Deﬁne the function f : X → N by f (abc) a + b + c, where a, b, and c are the digits of the number in X (write numbers less than 100 with leading 0’s to make them three digits). For example, f (253) 2 + 5 + 3 10. (a) Let A {n ∈ X : 113 ≤ n ≤ 122}. Find f (A). (b) Find f −1 ({1, 2}) (c) Find f −1 (3). (d) Find f −1 (28). (e) Is f injective? Explain. (f) Is f surjective? Explain. 15. Consider the set N2 N × N, the set of all ordered pairs (a, b) where a and b are natural numbers. Consider a function f : N2 → N given by f ((a, b)) a + b. (a) Let A {(a, b) ∈ N2 : a, b ≤ 10}. Find f (A). (b) Find f −1 (3) and f −1 ({0, 1, 2, 3}). (c) Give geometric descriptions of f −1 (n) and f −1 ({0, 1, . . . , n}) for any n ≥ 1. (d) Find f −1 (8) and f −1 ({0, 1, . . . , 8}) . 16. Let f : X → Y be some function. Suppose 3 ∈ Y. What can you say about f −1 (3) if you know, (a) f is injective? Explain. 0.4. Functions 55 (b) f is surjective? Explain. (c) f is bĳective? Explain. 17. Find a set X and a function f : X → N so that f −1 (0) ∪ f −1 (1) X. 18. What can you deduce about the sets X and Y if you know, (a) there is an injective function f : X → Y? Explain. (b) there is a surjective function f : X → Y? Explain. (c) there is a bĳective function f : X → Y? Explain. 19. Suppose f : X → Y is a function. Which of the following are possible? Explain. (a) f is injective but not surjective. (b) f is surjective but not injective. (c) |X| |Y| and f is injective but not surjective. (d) |X| |Y| and f is surjective but not injective. (e) |X| |Y|, X and Y are ﬁnite, and f is injective but not surjective. (f) |X| |Y|, X and Y are ﬁnite, and f is surjective but not injective. 20. Let f : X → Y and g : Y → Z be functions. We can deﬁne the composition of f and g to be the function g ◦ f : X → Z for which the image of each x ∈ X is g( f (x)). That is, plug x into f , then plug the result into g (just like composition in algebra and calculus). (a) If f and g are both injective, must g ◦ f be injective? Explain. (b) If f and g are both surjective, must g ◦ f be surjective? Explain. (c) Suppose g ◦ f is injective. What, if anything, can you say about f and g? Explain. (d) Suppose g ◦ f is surjective. What, if anything, can you say about f and g? Explain. ( 21. Consider the function f : Z → Z given by f (n) n+1 if n is even n−3 if n is odd. (a) Is f injective? Prove your answer. (b) Is f surjective? Prove your answer. 22. At the end of the semester a teacher assigns letter grades to each of her students. Is this a function? If so, what sets make up the domain and codomain, and is the function injective, surjective, bĳective, or neither? 56 0. Introduction and Preliminaries 23. In the game of Hearts, four players are each dealt 13 cards from a deck of 52. Is this a function? If so, what sets make up the domain and codomain, and is the function injective, surjective, bĳective, or neither? 24. Seven players are playing 5-card stud. Each player initially receives 5 cards from a deck of 52. Is this a function? If so, what sets make up the domain and codomain, and is the function injective, surjective, bĳective, or neither? 25. Consider the function f : N → N that gives the number of handshakes that take place in a room of n people assuming everyone shakes hands with everyone else. Give a recursive deﬁnition for this function. 26. Let f : X → Y be a function and A ⊆ X be a ﬁnite subset of the domain. What can you say about the relationship between |A| and f (A) ? Consider both the general case and what happens when you know f is injective, surjective, or bĳective. 27. Let f : X → Y be a function and B ⊆ Y be a ﬁnite subset of the codomain. What can you say about the relationship between |B| and f −1 (B) ? Consider both the general case and what happens when you know f is injective, surjective, or bĳective. 28. Let f : X → Y be a function, A ⊆ X and B ⊆ Y. (a) Is f −1 f (A) A? Always, sometimes, never? Explain. (b) Is f f −1 (B) B? Always, sometimes, never? Explain. (c) If one or both of the above do not always hold, is there something else you can say? Will equality always hold for particular types of functions? Is there some other relationship other than equality that would always hold? Explore. 29. Let f : X → Y be a function and A, B ⊆ X be subsets of the domain. (a) Is f (A ∪ B) f (A) ∪ f (B)? Always, sometimes, or never? Explain. (b) Is f (A ∩ B) f (A) ∩ f (B)? Always, sometimes, or never? Explain. 30. Let f : X → Y be a function and A, B ⊆ Y be subsets of the codomain. (a) Is f −1 (A ∪ B) f −1 (A) ∪ f −1 (B)? Always, sometimes, or never? Explain. (b) Is f −1 (A ∩ B) f −1 (A) ∩ f −1 (B)? Always, sometimes, or never? Explain. Chapter 1 Counting One of the ﬁrst things you learn in mathematics is how to count. Now we want to count large collections of things quickly and precisely. For example: • In a group of 10 people, if everyone shakes hands with everyone else exactly once, how many handshakes took place? • How many ways can you distribute 10 girl scout cookies to 7 boy scouts? • How many anagrams are there of “anagram”? Before tackling questions like these, let’s look at the basics of counting. 1.1 Additive and Multiplicative Principles Investigate! 1. A restaurant oﬀers 8 appetizers and 14 entrées. How many choices do you have if: (a) you will eat one dish, either an appetizer or an entrée? (b) you are extra hungry and want to eat both an appetizer and an entrée? 2. Think about the methods you used to solve question 1. Write down the rules for these methods. 3. Do your rules work? A standard deck of playing cards has 26 red cards and 12 face cards. (a) How many ways can you select a card which is either red or a face card? (b) How many ways can you select a card which is both red and a face card? (c) How many ways can you select two cards so that the ﬁrst one is red and the second one is a face card? ! Attempt the above activity before proceeding 57 ! 58 1. Counting Consider this rather simple counting problem: at Red Dogs and Donuts, there are 14 varieties of donuts, and 16 types of hot dogs. If you want either a donut or a dog, how many options do you have? This isn’t too hard, just add 14 and 16. Will that always work? What is important here? Additive Principle. The additive principle states that if event A can occur in m ways, and event B can occur in n disjoint ways, then the event “A or B” can occur in m + n ways. It is important that the events be disjoint: i.e., that there is no way for A and B to both happen at the same time. For example, a standard deck of 52 cards contains 26 red cards and 12 face cards. However, the number of ways to select a card which is either red or a face card is not 26 + 12 38. This is because there are 6 cards which are both red and face cards. Example 1.1.1 How many two letter “words” start with either A or B? (A word is just a string of letters; it doesn’t have to be English, or even pronounceable.) Solution. First, how many two letter words start with A? We just need to select the second letter, which can be accomplished in 26 ways. So there are 26 words starting with A. There are also 26 words that start with B. To select a word which starts with either A or B, we can pick the word from the ﬁrst 26 or the second 26, for a total of 52 words. The additive principle also works with more than two events. Say, in addition to your 14 choices for donuts and 16 for dogs, you would also consider eating one of 15 waﬄes? How many choices do you have now? You would have 14 + 16 + 15 45 options. Example 1.1.2 How many two letter words start with one of the 5 vowels? Solution. There are 26 two letter words starting with A, another 26 starting with E, and so on. We will have 5 groups of 26. So we add 26 to itself 5 times. Of course it would be easier to just multiply 5 · 26. We are really using the additive principle again, just using multiplication as a shortcut. 1.1. Additive and Multiplicative Principles 59 Example 1.1.3 Suppose you are going for some fro-yo. You can pick one of 6 yogurt choices, and one of 4 toppings. How many choices do you have? Solution. Break your choices up into disjoint events: A are the choices with the ﬁrst topping, B the choices featuring the second topping, and so on. There are four events; each can occur in 6 ways (one for each yogurt ﬂavor). The events are disjoint, so the total number of choices is 6 + 6 + 6 + 6 24. Note that in both of the previous examples, when using the additive principle on a bunch of events all the same size, it is quicker to multiply. This really is the same, and not just because 6 + 6 + 6 + 6 4 · 6. We can ﬁrst select the topping in 4 ways (that is, we ﬁrst select which of the disjoint events we will take). For each of those ﬁrst 4 choices, we now have 6 choices of yogurt. We have: Multiplicative Principle. The multiplicative principle states that if event A can occur in m ways, and each possibility for A allows for exactly n ways for event B, then the event “A and B” can occur in m · n ways. The multiplicative principle generalizes to more than two events as well. Example 1.1.4 How many license plates can you make out of three letters followed by three numerical digits? Solution. Here we have six events: the ﬁrst letter, the second letter, the third letter, the ﬁrst digit, the second digit, and the third digit. The ﬁrst three events can each happen in 26 ways; the last three can each happen in 10 ways. So the total number of license plates will be 26 · 26 · 26 · 10 · 10 · 10, using the multiplicative principle. Does this make sense? Think about how we would pick a license plate. How many choices we would have? First, we need to pick the ﬁrst letter. There are 26 choices. Now for each of those, there are 26 choices for the second letter: 26 second letters with ﬁrst letter A, 26 second letters with ﬁrst letter B, and so on. We add 26 60 1. Counting to itself 26 times. Or quicker: there are 26 · 26 choices for the ﬁrst two letters. Now for each choice of the ﬁrst two letters, we have 26 choices for the third letter. That is, 26 third letters for the ﬁrst two letters AA, 26 choices for the third letter after starting AB, and so on. There are 26 · 26 of these 26 third letter choices, for a total of (26 · 26) · 26 choices for the ﬁrst three letters. And for each of these 26 · 26 · 26 choices of letters, we have a bunch of choices for the remaining digits. In fact, there are going to be exactly 1000 choices for the numbers. We can see this because there are 1000 three-digit numbers (000 through 999). This is 10 choices for the ﬁrst digit, 10 for the second, and 10 for the third. The multiplicative principle says we multiply: 10 · 10 · 10 1000. All together, there were 263 choices for the three letters, and 103 choices for the numbers, so we have a total of 263 · 103 choices of license plates. Careful: “and” doesn’t mean “times.” For example, how many playing cards are both red and a face card? Not 26 · 12. The answer is 6, and we needed to know something about cards to answer that question. Another caution: how many ways can you select two cards, so that the ﬁrst one is a red card and the second one is a face card? This looks more like the multiplicative principle (you are counting two separate events) but the answer is not 26 · 12 here either. The problem is that while there are 26 ways for the ﬁrst card to be selected, it is not the case that for each of those there are 12 ways to select the second card. If the ﬁrst card was both red and a face card, then there would be only 11 choices for the second card.1 Example 1.1.5 Counting functions. How many functions f : {1, 2, 3, 4, 5} → {a, b, c, d} are there? Solution. Remember that a function sends each element of the domain to exactly one element of the codomain. To determine a function, we just need to specify the image of each element in the domain. Where can we send 1? There are 4 choices. Where can we send 2? Again, 4 choices. What we have here is 5 “events” (picking 1To solve this problem, you could break it into two cases. First, count how many ways there are to select the two cards when the ﬁrst card is a red non-face card. Second, count how many ways when the ﬁrst card is a red face card. Doing so makes the events in each separate case independent, so the multiplicative principle can be applied. 1.1. Additive and Multiplicative Principles 61 the image of an element in the domain) each of which can happen in 4 ways (the choices for that image). Thus there are 4·4·4·4·4 45 functions. This is more than just an example of how we can use the multiplicative principle in a particular counting question. What we have here is a general interpretation of certain applications of the multiplicative principle using rigorously deﬁned mathematical objects: functions. Whenever we have a counting question that asks for the the number of outcomes of a repeated event, we can interpret that as asking for the number of functions from {1, 2, . . . , n} (where n is the number of times the event is repeated) to {1, 2, . . . , k} (where k is the number of ways that event can occur). Counting With Sets Do you believe the additive and multiplicative principles? How would you convince someone they are correct? This is surprisingly diﬃcult. They seem so simple, so obvious. But why do they work? To make things clearer, and more mathematically rigorous, we will use sets. Do not skip this section! It might seem like we are just trying to give a proof of these principles, but we are doing a lot more. If we understand the additive and multiplicative principles rigorously, we will be better at applying them, and knowing when and when not to apply them at all. We will look at the additive and multiplicative principles in a slightly diﬀerent way. Instead of thinking about event A and event B, we want to think of a set A and a set B. The sets will contain all the diﬀerent ways the event can happen. (It will be helpful to be able to switch back and forth between these two models when checking that we have counted correctly.) Here’s what we mean: Example 1.1.6 Suppose you own 9 shirts and 5 pairs of pants. 1. How many outﬁts can you make? 2. If today is half-naked-day, and you will wear only a shirt or only a pair of pants, how many choices do you have? Solution. By now you should agree that the answer to the ﬁrst question is 9 · 5 45 and the answer to the second question is 9 + 5 14. These are the multiplicative and additive principles. 62 1. Counting There are two events: picking a shirt and picking a pair of pants. The ﬁrst event can happen in 9 ways and the second event can happen in 5 ways. To get both a shirt and a pair of pants, you multiply. To get just one article of clothing, you add. Now look at this using sets. There are two sets, call them S and P. The set S contains all 9 shirts so |S| 9 while |P| 5, since there are 5 elements in the set P (namely your 5 pairs of pants). What are we asking in terms of these sets? Well in question 2, we really want |S ∪ P|, the number of elements in the union of shirts and pants. This is just |S| + |P| (since there is no overlap; |S ∩ P| 0). Question 1 is slightly more complicated. Your ﬁrst guess might be to ﬁnd |S ∩ P|, but this is not right (there is nothing in the intersection). We are not asking for how many clothing items are both a shirt and a pair of pants. Instead, we want one of each. We could think of this as asking how many pairs (x, y) there are, where x is a shirt and y is a pair of pants. As we will soon verify, this number is |S| · |P|. From this example we can see right away how to rephrase our additive principle in terms of sets: Additive Principle (with sets). Given two sets A and B, if A ∩ B ∅ (that is, if there is no element in common to both A and B), then |A ∪ B| |A| + |B| . This hardly needs a proof. To ﬁnd A ∪ B, you take everything in A and throw in everything in B. Since there is no element in both sets already, you will have |A| things and add |B| new things to it. This is what adding does! Of course, we can easily extend this to any number of disjoint sets. From the example above, we see that in order to investigate the multiplicative principle carefully, we need to consider ordered pairs. We should deﬁne this carefully: Cartesian Product. Given sets A and B, we can form the set A × B {(x, y) : x ∈ A ∧ y ∈ B} to be the set of all ordered pairs (x, y) where x is an element of A and y is an element of B. We call A × B the Cartesian product of A and B. 1.1. Additive and Multiplicative Principles 63 Example 1.1.7 Let A {1, 2} and B {3, 4, 5}. Find A × B. Solution. We want to ﬁnd ordered pairs (a, b) where a can be either 1 or 2 and b can be either 3, 4, or 5. A × B is the set of all of these pairs: A × B {(1, 3), (1, 4), (1, 5), (2, 3), (2, 4), (2, 5)}. The question is, what is |A × B|? To ﬁgure this out, write out A × B. Let A {a 1 , a 2 , a3 , . . . , a m } and B {b1 , b 2 , b3 , . . . , b n } (so |A| m and |B| n). The set A × B contains all pairs with the ﬁrst half of the pair being some a i ∈ A and the second being one of the b j ∈ B. In other words: A × B {(a 1 , b1 ), (a 1 , b2 ), (a 1 , b3 ), . . . (a 1 , b n ), (a 2 , b1 ), (a2 , b2 ), (a 2 , b3 ), . . . , (a 2 , b n ), (a 3 , b1 ), (a3 , b2 ), (a 3 , b3 ), . . . , (a 3 , b n ), .. . (a m , b1 ), (a m , b 2 ), (a m , b 3 ), . . . , (a m , b n )}. Notice what we have done here: we made m rows of n pairs, for a total of m · n pairs. Each row above is really {a i } × B for some a i ∈ A. That is, we ﬁxed the A-element. Broken up this way, we have A × B ({a 1 } × B) ∪ ({a 2 } × B) ∪ ({a 3 } × B) ∪ · · · ∪ ({a m } × B). So A × B is really the union of m disjoint sets. Each of those sets has n elements in them. The total (using the additive principle) is n + n + n + · · · + n m · n. To summarize: Multiplicative Principle (with sets). Given two sets A and B, we have |A × B| |A| · |B|. Again, we can easily extend this to any number of sets. 64 1. Counting Principle of Inclusion/Exclusion Investigate! A recent buzz marketing campaign for Village Inn surveyed patrons on their pie preferences. People were asked whether they enjoyed (A) Apple, (B) Blueberry or (C) Cherry pie (respondents answered yes or no to each type of pie, and could say yes to more than one type). The following table shows the results of the survey. Pies enjoyed: A B C AB AC BC ABC Number of people: 20 13 26 9 15 7 5 How many of those asked enjoy at least one of the kinds of pie? Also, explain why the answer is not 95. ! Attempt the above activity before proceeding ! While we are thinking about sets, consider what happens to the additive principle when the sets are NOT disjoint. Suppose we want to ﬁnd |A ∪ B| and know that |A| 10 and |B| 8. This is not enough information though. We do not know how many of the 8 elements in B are also elements of A. However, if we also know that |A ∩ B| 6, then we can say exactly how many elements are in A, and, of those, how many are in B and how many are not (6 of the 10 elements are in B, so 4 are in A but not in B). We could ﬁll in a Venn diagram as follows: A B 4 6 2 This says there are 6 elements in A ∩ B, 4 elements in A \ B and 2 elements in B \ A. Now these three sets are disjoint, so we can use the additive principle to ﬁnd the number of elements in A ∪ B. It is 6 + 4 + 2 12. This will always work, but drawing a Venn diagram is more than we need to do. In fact, it would be nice to relate this problem to the case where A and B are disjoint. Is there one rule we can make that works in either case? Here is another way to get the answer to the problem above. Start by just adding |A| + |B|. This is 10 + 8 18, which would be the answer if |A ∩ B| 0. We see that we are oﬀ by exactly 6, which just so happens to 1.1. Additive and Multiplicative Principles 65 be |A ∩ B|. So perhaps we guess, |A ∪ B| |A| + |B| − |A ∩ B| . This works for this one example. Will it always work? Think about what we are doing here. We want to know how many things are either in A or B (or both). We can throw in everything in A, and everything in B. This would give |A| + |B| many elements. But of course when you actually take the union, you do not repeat elements that are in both. So far we have counted every element in A ∩ B exactly twice: once when we put in the elements from A and once when we included the elements from B. We correct by subtracting out the number of elements we have counted twice. So we added them in twice, subtracted once, leaving them counted only one time. In other words, we have: Cardinality of a union (2 sets). For any ﬁnite sets A and B, |A ∪ B| |A| + |B| − |A ∩ B| . We can do something similar with three sets. Example 1.1.8 An examination in three subjects, Algebra, Biology, and Chemistry, was taken by 41 students. The following table shows how many students failed in each single subject and in their various combinations: Subject: A B C AB AC BC ABC Failed: 12 5 8 2 6 3 1 How many students failed at least one subject? Solution. The answer is not 37, even though the sum of the numbers above is 37. For example, while 12 students failed Algebra, 2 of those students also failed Biology, 6 also failed Chemestry, and 1 of those failed all three subjects. In fact, that 1 student who failed all three subjects is counted a total of 7 times in the total 37. To clarify things, let us think of the students who failed Algebra as the elements of the set A, and similarly for sets B and C. The one student who failed all three subjects is the lone element of the set A ∩ B ∩ C. Thus, in Venn diagrams: 66 1. Counting A B 1 C Now let’s ﬁll in the other intersections. We know A ∩ B contains 2 elements, but 1 element has already been counted. So we should put a 1 in the region where A and B intersect (but C does not). Similarly, we calculate the cardinality of (A ∩ C) \ B, and (B ∩ C) \ A: A B 1 5 1 2 C Next, we determine the numbers which should go in the remaining regions, including outside of all three circles. This last number is the number of students who did not fail any subject: A B 1 5 5 26 1 1 2 0 C We found 5 goes in the “A only” region because the entire circle for A needed to have a total of 12, and 7 were already accounted for. Similarly, we calculate the “B only” region to contain only 1 student and the “C only” region to contain no students. Thus the number of students who failed at least one class is 15 (the sum of the numbers in each of the eight disjoint regions). The number of students who passed all three classes is 26: the total number of students, 41, less the 15 who failed at least one class. Note that we can also answer other questions. For example, how many students failed just Chemistry? None. How many passed Algebra but failed both Biology and Chemistry? This corresponds to the region inside both B and C but outside of A, containing 2 students. Could we have solved the problem above in an algebraic way? While the additive principle generalizes to any number of sets, when we add a third set here, we must be careful. With two sets, we needed to know the 1.1. Additive and Multiplicative Principles 67 cardinalities of A, B, and A ∩ B in order to ﬁnd the cardinality of A ∪ B. With three sets we need more information. There are more ways the sets can combine. Not surprisingly then, the formula for cardinality of the union of three non-disjoint sets is more complicated: Cardinality of a union (3 sets). For any ﬁnite sets A, B, and C, |A ∪ B ∪ C| |A|+|B|+|C|−|A ∩ B|−|A ∩ C|−|B ∩ C|+|A ∩ B ∩ C| . To determine how many elements are in at least one of A, B, or C we add up all the elements in each of those sets. However, when we do that, any element in both A and B is counted twice. Also, each element in both A and C is counted twice, as are elements in B and C, so we take each of those out of our sum once. But now what about the elements which are in A ∩ B ∩ C (in all three sets)? We added them in three times, but also removed them three times. They have not yet been counted. Thus we add those elements back in at the end. Returning to our example above, we have |A| 12, |B| 5, |C| 8. We also have |A ∩ B| 2, |A ∩ C| 6, |B ∩ C| 3, and |A ∩ B ∩ C| 1. Therefore: |A ∪ B ∪ C| 12 + 5 + 8 − 2 − 6 − 3 + 1 15. This is what we got when we solved the problem using Venn diagrams. This process of adding in, then taking out, then adding back in, and so on is called the Principle of Inclusion/Exclusion, or simply PIE. We will return to this counting technique later to solve for more complicated problems (involving more than 3 sets). Exercises 1. Your wardrobe consists of 5 shirts, 3 pairs of pants, and 17 bow ties. bow ties How many diﬀerent outﬁts can you make? 2. For your college interview, you must wear a tie. You own 3 regular (boring) ties and 5 (cool) bow ties. (a) How many choices do you have for your neck-wear? (b) You realize that the interview is for clown college, so you should probably wear both a regular tie and a bow tie. How many choices do you have now? (c) For the rest of your outﬁt, you have 5 shirts, 4 skirts, 3 pants, and 7 dresses. You want to select either a shirt to wear with a skirt or pants, or just a dress. How many outﬁts do you have to choose from? 68 1. Counting 3. Your Blu-ray collection consists of 9 comedies and 7 horror movies. Give an example of a question for which the answer is: (a) 16. (b) 63. 4. We usually write numbers in decimal form (or base 10), meaning numbers are composed using 10 diﬀerent “digits” {0, 1, . . . , 9}. Sometimes though it is useful to write numbers hexadecimal or base 16. Now there are 16 distinct digits that can be used to form numbers: {0, 1, . . . , 9, A, B, C, D, E, F}. So for example, a 3 digit hexadecimal number might be 2B8. (a) How many 2-digit hexadecimals are there in which the ﬁrst digit is E or F? Explain your answer in terms of the additive principle (using either events or sets). (b) Explain why your answer to the previous part is correct in terms of the multiplicative principle (using either events or sets). Why do both the additive and multiplicative principles give you the same answer? (c) How many 3-digit hexadecimals start with a letter (A-F) and end with a numeral (0-9)? Explain. (d) How many 3-digit hexadecimals start with a letter (A-F) or end with a numeral (0-9) (or both)? Explain. 5. Suppose you have sets A and B with |A| 10 and |B| 15. (a) What is the largest possible value for |A ∩ B| ? (b) What is the smallest possible value for |A ∩ B| ? (c) What are the possible values for |A ∪ B| ? 6. If |A| 8 and |B| 5, what is |A ∪ B| + |A ∩ B| ? 7. A group of college students were asked about their TV watching habits. Of those surveyed, 28 students watch The Walking Dead, 19 watch The Blacklist, and 24 watch Game of Thrones. Additionally, 16 watch The Walking Dead and The Blacklist, 14 watch The Walking Dead and Game of Thrones, and 10 watch The Blacklist and Game of Thrones. There are 8 students who watch all three shows. How many students surveyed watched at least one of the shows? 8. In a recent survey, 30 students reported whether they liked their potatoes Mashed, French-fried, or Twice-baked. 15 liked them mashed, 20 liked French fries, and 9 liked twice baked potatoes. Additionally, 12 students liked both mashed and fried potatoes, 5 liked French fries 1.1. Additive and Multiplicative Principles 69 and twice baked potatoes, 6 liked mashed and baked, and 3 liked all three styles. How many students hate potatoes? Explain why your answer is correct. 9. For how many n ∈ {1, 2, . . . , 500} is n a multiple of one or more of 5, 6, or 7? 10. How many positive integers less than 1000 are multiples of 3, 5, or 7? Explain your answer using the Principle of Inclusion/Exclusion. 11. Let A, B, and C be sets. (a) Find |(A ∪ C) \ B| provided |A| 50, |B| 45, |C| 40, |A ∩ B| 20, |A ∩ C| 15, |B ∩ C| 23, and |A ∩ B ∩ C| 12. (b) Describe a set in terms of A, B, and C with cardinality 26. 12. Consider all 5 letter “words” made from the letters a through h. (Recall, words are just strings of letters, not necessarily actual English words.) (a) How many of these words are there total? (b) How many of these words contain no repeated letters? (c) How many of these words start with the sub-word “aha”? (d) How many of these words either start with “aha” or end with “bah” or both? (e) How many of the words containing no repeats also do not contain the sub-word “bad”? 13. For how many three digit numbers (100 to 999) is the sum of the digits even? (For example, 343 has an even sum of digits: 3 + 4 + 3 10 which is even.) Find the answer and explain why it is correct in at least two diﬀerent ways. 14. The number 735000 factors as 23 · 3 · 54 · 72 . How many divisors does it have? Explain your answer using the multiplicative principle. 70 1. Counting 1.2 Binomial Coefficients Investigate! In chess, a rook can move only in straight lines (not diagonally). Fill in each square of the chess board below with the number of diﬀerent shortest paths the rook, in the upper left corner, can take to get to that square. For example, one square is already ﬁlled in. There are six diﬀerent paths from the rook to the square: DDRR (down down right right), DRDR, DRRD, RDDR, RDRD and RRDD. R 6 ! Attempt the above activity before proceeding ! Here are some apparently diﬀerent discrete objects we can count: subsets, bit strings, lattice paths, and binomial coeﬃcients. We will give an example of each type of counting problem (and say what these things even are). As we will see, these counting problems are surprisingly similar. Subsets Subsets should be familiar, otherwise read over Section 0.3 again. Suppose we look at the set A {1, 2, 3, 4, 5}. How many subsets of A contain exactly 3 elements? First, a simpler question: How many subsets of A are there total? In other words, what is |P(A)| (the cardinality of the power set of A)? Think about how we would build a subset. We need to decide, for each of the elements of A, whether or not to include the element in our subset. So we need to decide “yes” or “no” for the element 1. And for each choice we make, we need to decide “yes” or “no” for the element 2. And so on. 1.2. Binomial Coefficients 71 For each of the 5 elements, we have 2 choices. Therefore the number of subsets is simply 2 · 2 · 2 · 2 · 2 25 (by the multiplicative principle). Of those 32 subsets, how many have 3 elements? This is not obvious. Note that we cannot just use the multiplicative principle. Maybe we want to say we have 2 choices (yes/no) for the ﬁrst element, 2 choices for the second, 2 choices for the third, and then only 1 choice for the other two. But what if we said “no” to one of the ﬁrst three elements? Then we would have two choices for the 4th element. What a mess! Another (bad) idea: we need to pick three elements to be in our subset. There are 5 elements to choose from. So there are 5 choices for the ﬁrst element, and for each of those 4 choices for the second, and then 3 for the third (last) element. The multiplicative principle would say then that there are a total of 5 · 4 · 3 60 ways to select the 3-element subset. But this cannot be correct (60 > 32 for one thing). One of the outcomes we would get from these choices would be the set {3, 2, 5}, by choosing the element 3 ﬁrst, then the element 2, then the element 5. Another outcome would be {5, 2, 3} by choosing the element 5 ﬁrst, then the element 2, then the element 3. But these are the same set! We can correct this by dividing: for each set of three elements, there are 6 outcomes counted among our 60 (since there are 3 choices for which element we list ﬁrst, 2 for which we list second, and 1 for which we list last). So we expect there to be 10 3-element subsets of A. Is this right? Well, we could list out all 10 of them, being very systematic in doing so, to make sure we don’t miss any or list any twice. Or we could try to count how many subsets of A don’t have 3 elements in them. How many have no elements? Just 1 (the empty set). How many have 5? Again, just 1. These are the cases in which we say “no” to all elements, or “yes” to all elements. Okay, what about the subsets which contain a single element? There are 5 of these. We must say “yes” to exactly one element, and there are 5 to choose from. This is also the number of subsets containing 4 elements. Those are the ones for which we must say “no” to exactly one element. So far we have counted 12 of the 32 subsets. We have not yet counted the subsets with cardinality 2 and with cardinality 3. There are a total of 20 subsets left to split up between these two groups. But the number of each must be the same! If we say “yes” to exactly two elements, that can be accomplished in exactly the same number of ways as the number of ways we can say “no” to exactly two elements. So the number of 2-element subsets is equal to the number of 3-element subsets. Together there are 20 of these subsets, so 10 each. Number of elements: Number of subsets: 0 1 1 5 2 10 3 10 4 5 5 1 72 1. Counting Bit Strings “Bit” is short for “binary digit,” so a bit string is a string of binary digits. The binary digits are simply the numbers 0 and 1. All of the following are bit strings: 1001 0 1111 1010101010. The number of bits (0’s or 1’s) in the string is the length of the string; the strings above have lengths 4, 1, 4, and 10 respectively. We also can ask how many of the bits are 1’s. The number of 1’s in a bit string is the weight of the string; the weights of the above strings are 2, 0, 4, and 5 respectively. Bit Strings. • An n-bit string is a bit string of length n. That is, it is a string containing n symbols, each of which is a bit, either 0 or 1. • The weight of a bit string is the number of 1’s in it. • Bn is the set of all n-bit strings. • Bnk is the set of all n-bit strings of weight k. For example, the elements of the set B32 are the bit strings 011, 101, and 110. Those are the only strings containing three bits exactly two of which are 1’s. The counting questions: How many bit strings have length 5? How many of those have weight 3? In other words, we are asking for the cardinalities |B5 | and |B53 |. To ﬁnd the number of 5-bit strings is straight forward. We have 5 bits, and each can either be a 0 or a 1. So there are 2 choices for the ﬁrst bit, 2 choices for the second, and so on. By the multiplicative principle, there are 2 · 2 · 2 · 2 · 2 25 32 such strings. Finding the number of 5-bit strings of weight 3 is harder. Think about how such a string could start. The ﬁrst bit must be either a 0 or a 1. In the ﬁrst case (the string starts with a 0), we must then decide on four more bits. To have a total of three 1’s, among those four remaining bits there must be three 1’s. To count all of these strings, we must include all 4-bit strings of weight 3. In the second case (the string starts with a 1), we still have four bits to choose, but now only two of them can be 1’s, so we should look at all the 4-bit strings of weight 2. So the strings in B53 all have the form 1B42 (that is, a 1 followed by a string from B42 ) or 0B43 . These two sets are disjoint, so we can use the additive principle: |B53 | |B42 | + |B43 |. 1.2. Binomial Coefficients 73 This is an example of a recurrence relation. We represented one instance of our counting problem in terms of two simpler instances of the problem. If only we knew the cardinalities of B42 and B43 . Repeating the same reasoning, |B42 | |B31 | + |B32 | and |B43 | |B32 | + |B33 |. We can keep going down, but this should be good enough. Both B31 and B32 contain 3 bit strings: we must pick one of the three bits to be a 1 (three ways to do that) or one of the three bits to be a 0 (three ways to do that). Also, B33 contains just one string: 111. Thus |B42 | 6 and |B43 | 4, which puts B53 at a total of 10 strings. But wait —32 and 10 were the answers to the counting questions about subsets. Coincidence? Not at all. Each bit string can be thought of as a code for a subset. To represent the subsets of A {1, 2, 3, 4, 5}, we can use 5-bit strings, one bit for each element of A. Each bit in the string is a 0 if its corresponding element of A is not in the subset, and a 1 if the element of A is in the subset. Remember, deciding the subset amounted to a sequence of ﬁve yes/no votes for the elements of A. Instead of yes, we put a 1; instead of no, we put a 0. For example, the bit string 11001 represents the subset {1, 2, 5} since the ﬁrst, second and ﬁfth bits are 1’s. The subset {3, 5} would be coded by the string 00101. What we really have here is a bĳection from P(A) to B5 . Now for a subset to contain exactly three elements, the corresponding bit string must contain exactly three 1’s. In other words, the weight must be 3. Thus counting the number of 3-element subsets of A is the same as counting the number 5-bit strings of weight 3. Lattice Paths The integer lattice is the set of all points in the Cartesian plane for which both the x and y coordinates are integers. If you like to draw graphs on graph paper, the lattice is the set of all the intersections of the grid lines. A lattice path is one of the shortest possible paths connecting two points on the lattice, moving only horizontally and vertically. For example, here are three possible lattice paths from the point (0, 0) to (3, 2): (3,2) (0,0) (3,2) (0,0) (3,2) (0,0) Notice to ensure the path is the shortest possible, each move must be either to the right or up. Additionally, in this case, note that no matter 74 1. Counting what path we take, we must make three steps right and two steps up. No matter what order we make these steps, there will always be 5 steps. Thus each path has length 5. The counting question: how many lattice paths are there between (0, 0) and (3, 2)? We could try to draw all of these, or instead of drawing them, maybe just list which direction we travel on each of the 5 steps. One path might be RRUUR, or maybe UURRR, or perhaps RURRU (those correspond to the three paths drawn above). So how many such strings of R’s and U’s are there? Notice that each of these strings must contain 5 symbols. Exactly 3 of them must be R’s (since our destination is 3 units to the right). This seems awfully familiar. In fact, what if we used 1’s instead of R’s and 0’s instead of U’s? Then we would just have 5-bit strings of weight 3. There are 10 of those, so there are 10 lattice paths from (0,0) to (3,2). The correspondence between bit strings and lattice paths does not stop there. Here is another way to count lattice paths. Consider the lattice shown below: A (3,2) B (0,0) Any lattice path from (0,0) to (3,2) must pass through exactly one of A and B. The point A is 4 steps away from (0,0) and two of them are towards the right. The number of lattice paths to A is the same as the number of 4-bit strings of weight 2, namely 6. The point B is 4 steps away from (0,0), but now 3 of them are towards the right. So the number of paths to point B is the same as the number of 4-bit strings of weight 3, namely 4. So the total number of paths to (3,2) is just 6 + 4. This is the same way we calculated the number of 5-bit strings of weight 3. The point: the exact same recurrence relation exists for bit strings and for lattice paths. Binomial Coefficients Binomial coeﬃcients are the coeﬃcients in the expanded version of a binomial, such as (x + y)5 . What happens when we multiply such a binomial out? We will expand (x + y)n for various values of n. Each of these are done by multiplying everything out (i.e., FOIL-ing) and then collecting like terms. (x + y)1 x + y (x + y)2 x 2 + 2x y + y 2 (x + y)3 x 3 + 3x 2 y + 3x y 2 + y 3 1.2. Binomial Coefficients 75 (x + y)4 x 4 + 4x 3 y + 6x 2 y 2 + 4x y 3 + y 4 . In fact, there is a quicker way to expand the above binomials. For example, consider the next one, (x + y)5 . What we are really doing is multiplying out, (x + y)(x + y)(x + y)(x + y)(x + y). If that looks daunting, go back to the case of (x + y)3 (x + y)(x + y)(x + y). Why do we only have one x 3 and y 3 but three x 2 y and x y 2 terms? Every time we distribute over an (x + y) we create two copies of what is left, one multiplied by x, the other multiplied by y. To get x 3 , we need to pick the “multiplied by x” side every time (we don’t have any y’s in the term). This will only happen once. On the other hand, to get x 2 y we need to select the x side twice and the y side once. In other words, we need to pick one of the three (x + y) terms to “contribute” their y. Similarly, in the expansion of (x + y)5 , there will be only one x 5 term and one y 5 term. This is because to get an x 5 , we need to use the x term in each of the copies of the binomial (x + y), and similarly for y 5 . What about x 4 y? To get terms like this, we need to use four x’s and one y, so we need exactly one of the ﬁve binomials to contribute a y. There are 5 choices for this, so there are 5 ways to get x 4 y, so the coeﬃcient of x 4 y is 5. This is also the coeﬃcient for x y 4 for the same (but opposite) reason: there are 5 ways to pick which of the 5 binomials contribute the single x. So far we have (x + y)5 x 5 + 5x 4 y + ? x 3 y 2 + ? x 2 y 3 + 5x y 4 + y 5 . We still need the coeﬃcients of x 3 y 2 and x 2 y 3 . In both cases, we need to pick exactly 3 of the 5 binomials to contribute one variable, the other two to contribute the other. Wait. This sounds familiar. We have 5 things, each can be one of two things, and we need a total of 3 of one of them. That’s just like taking 5 bits and making sure exactly 3 of them are 1’s. So the coeﬃcient of x 3 y 2 (and also x 2 y 3 ) will be exactly the same as the number of bit strings of length 5 and weight 3, which we found earlier to be 10. So we have: (x + y)5 x 5 + 5x 4 y + 10x 3 y 2 + 10x 2 y 3 + 5x y 4 + y 5 . These numbers we keep seeing over and over again. They are the number of subsets of a particular size, the number of bit strings of a particular weight, the number of lattice paths, and the coeﬃcients of these binomial products. We will call them binomial coeﬃcients. We n even have a special symbol for them: k . 76 1. Counting Binomial Coeﬃcients. For each integer n ≥ 0 and integer k with 0 ≤ k ≤ n there is a number n , k read “n choose k.” We have: • • • • • n k n k |Bnk |, the number of n-bit strings of weight k. is the number of subsets of a set of size n each with cardinality k. n k is the number of lattice paths of length n containing k steps to the right. n k n k is the coeﬃcient of x k y n−k in the expansion of (x + y)n . is the number of ways to select k objects from a total of n objects. The last bullet point is usually taken as the deﬁnition of nk . Out of n objects we must choose k of them, so there are n choose k ways of doing this. Each of our counting problems above can be viewed in this way: • How many subsets of {1, 2, 3, 4, 5} contain exactly 3 elements? We must choose 3 of the 5 elements to be in our subset. There are 53 ways to do this, so there are 53 such subsets. • How many bit strings have length 5 and weight 3? We must choose 3 of the 5 bits to be 1’s. There are 53 ways to do this, so there are 53 such bit strings. • How many lattice paths are there from (0,0) to (3,2)? We must choose 3 of the 5 steps to be towards the right. There are 53 ways to do this, so there are 53 such lattice paths. • What is the coeﬃcient of x 3 y 2 in the expansion of (x + y)5 ? We must choose 3 of the 5 copies of the binomial to contribute an x. There 5 5 are 3 ways to do this, so the coeﬃcient is 3 . It should be clear that in each case above, we have the right answer. All we had to do is phrase the question correctly and it became obvious that 53 is correct. However, this does not tell us that the answer is in fact n 10 in each case. We will eventually ﬁnd a formula for k , but for now, look back at how we arrived at the answer 10 in our counting problems above. It all came down to bit strings, and we have a recurrence relation for bit strings: n−1 |Bnk | |Bn−1 k−1 | + |Bk |. 1.2. Binomial Coefficients 77 Remember, this is because we can start the bit string with either a 1 or a 0. In both cases, we have n − 1 more bits to pick. The strings starting with 1 must contain k − 1 more 1’s, while the strings starting with 0 still need k more 1’s. Since |Bnk | nk , the same recurrence relation holds for binomial coeﬃcients: Recurrence relation for n k . n n−1 n−1 + . k k−1 k Pascal’s Triangle n Let’s arrange the binomial coeﬃcients 4 0 3 0 2 0 4 1 1 0 3 1 0 0 2 1 4 2 k into a triangle like follows: 1 1 3 2 2 2 4 3 3 3 4 4 This can continue as far down as we like. The recurrence relation for tells us that each entry in the triangle is the sum of the two entries above it. The entries on the sides of the triangle are always 1. This is n because 0 1 for all n since there is only one way to pick 0 of n objects and nn 1 since there is one way to select all n out of n objects. Using the recurrence relation, and the fact that the sides of the triangle are 1’s, we can easily replace all the entries above with the correct values of nk . Doing so gives us Pascal’s triangle. We can use Pascal’s triangle to calculate binomial coeﬃcients. For example, using the triangle below, we can ﬁnd 12 924. 6 n k 78 1. Counting Pascal’s Triangle 1 1 2 1 3 1 1 5 7 1 9 1 1 1 1 14 15 1 16 66 78 13 91 45 55 11 12 1 36 10 1 1 28 1 5 10 35 56 84 1 4 20 15 21 8 1 3 10 6 1 1 6 4 1 1 6 15 35 70 1 1 7 21 56 126 126 1 28 84 8 1 36 120 210 252 210 120 9 1 45 165 330 462 462 330 165 10 55 220 495 792 924 792 495 220 1 11 66 286 715 1287 1716 1716 1287 715 286 1 12 78 364 1001 2002 3003 3432 3003 2002 1001 364 1 13 91 1 14 105 455 1365 3003 5005 6435 6435 5005 3003 1365 455 105 120 560 1820 4368 8008 11440 12870 11440 8008 4368 1820 1 15 560 120 1 16 1 Exercises 1. Let S {1, 2, 3, 4, 5, 6} (a) How many subsets are there total? (b) How many subsets have {2, 3, 5} as a subset? (c) How many subsets contain at least one odd number? (d) How many subsets contain exactly one even number? 2. Let S {1, 2, 3, 4, 5, 6} (a) How many subsets are there of cardinality 4? (b) How many subsets of cardinality 4 have {2, 3, 5} as a subset? (c) How many subsets of cardinality 4 contain at least one odd number? (d) How many subsets of cardinality 4 contain exactly one even number? 1.2. Binomial Coefficients 3. 79 Let A {1, 2, 3, . . . , 9}. (a) How many subsets of A are there? That is, ﬁnd |P(A)|. Explain. (b) How many subsets of A contain exactly 5 elements? Explain. (c) How many subsets of A contain only even numbers? Explain. (d) How many subsets of A contain an even number of elements? Explain. 4. How many 9-bit strings (that is, bit strings of length 9) are there which: (a) Start with the sub-string 101? Explain. (b) Have weight 5 (i.e., contain exactly ﬁve 1’s) and start with the sub-string 101? Explain. (c) Either start with 101 or end with 11 (or both)? Explain. (d) Have weight 5 and either start with 101 or end with 11 (or both)? Explain. 5. You break your piggy-bank to discover lots of pennies and nickels. You start arranging these in rows of 6 coins. (a) You ﬁnd yourself making rows containing an equal number of pennies and nickels. For fun, you decide to lay out every possible such row. How many coins will you need? (b) How many coins would you need to make all possible rows of 6 coins (not necessarily with equal number of pennies and nickels)? 6. How many 10-bit strings contain 6 or more 1’s? 7. How many subsets of {0, 1, . . . , 9} have cardinality 6 or more? 8. What is the coeﬃcient of x 12 in (x + 2)15 ? 9. What is the coeﬃcient of x 9 in the expansion of (x + 1)14 + x 3 (x + 2)15 ? 10. How many lattice paths start at (3,3) and (a) end at (10,10)? (b) end at (10,10) and pass through (5,7)? (c) end at (10,10) and avoid (5,7)? 11. Gridtown USA, besides having excellent donut shops, is known for its precisely laid out grid of streets and avenues. Streets run east-west, 80 1. Counting and avenues north-south, for the entire stretch of the town, never curving and never interrupted by parks or schools or the like. Suppose you live on the corner of 3rd and 3rd and work on the corner of 12th and 12th. Thus you must travel 18 blocks to get to work as quickly as possible. (a) How many diﬀerent routes can you take to work, assuming you want to get there as quickly as possible? Explain. (b) Now suppose you want to stop and get a donut on the way to work, from your favorite donut shop on the corner of 10th ave and 8th st. How many routes to work, stopping at the donut shop, can you take (again, ensuring the shortest possible route)? Explain. (c) Disaster Strikes Gridtown: there is a pothole on 4th ave between 5th st and 6th st. How many routes to work can you take avoiding that unsightly (and dangerous) stretch of road? Explain. (d) The pothole has been repaired (phew) and a new donut shop has opened on the corner of 4th ave and 5th st. How many routes to work drive by one or the other (or both) donut shops? Hint: the donut shops serve PIE. 12. Suppose you are ordering a large pizza from D.P. Dough. You want 3 distinct toppings, chosen from their list of 11 vegetarian toppings. (a) How many choices do you have for your pizza? (b) How many choices do you have for your pizza if you refuse to have pineapple as one of your toppings? (c) How many choices do you have for your pizza if you insist on having pineapple as one of your toppings? (d) How do the three questions above relate to each other? Explain. 13. Explain why the coeﬃcient of x 5 y 3 the same as the coeﬃcient of x 3 y 5 in the expansion of (x + y)8 ? 1.3. Combinations and Permutations 1.3 81 Combinations and Permutations Investigate! You have a bunch of chips which come in ﬁve diﬀerent colors: red, blue, green, purple and yellow. 1. How many diﬀerent two-chip stacks can you make if the bottom chip must be red or blue? Explain your answer using both the additive and multiplicative principles. 2. How many diﬀerent three-chip stacks can you make if the bottom chip must be red or blue and the top chip must be green, purple or yellow? How does this problem relate to the previous one? 3. How many diﬀerent three-chip stacks are there in which no color is repeated? What about four-chip stacks? 4. Suppose you wanted to take three diﬀerent colored chips and put them in your pocket. How many diﬀerent choices do you have? What if you wanted four diﬀerent colored chips? How do these problems relate to the previous one? ! Attempt the above activity before proceeding ! A permutation is a (possible) rearrangement of objects. For example, there are 6 permutations of the letters a, b, c: abc, acb, bac, bca, cab, cba. We know that we have them all listed above —there are 3 choices for which letter we put ﬁrst, then 2 choices for which letter comes next, which leaves only 1 choice for the last letter. The multiplicative principle says we multiply 3 · 2 · 1. Example 1.3.1 How many permutations are there of the letters a, b, c, d, e, f ? Solution. We do NOT want to try to list all of these out. However, if we did, we would need to pick a letter to write down ﬁrst. There are 6 choices for that letter. For each choice of ﬁrst letter, there are 5 choices for the second letter (we cannot repeat the ﬁrst letter; we are rearranging letters and only have one of each), and for each of those, there are 4 choices for the third, 3 choices for the fourth, 2 choices for the ﬁfth and ﬁnally only 1 choice for the last letter. So there are 6 · 5 · 4 · 3 · 2 · 1 720 permutations of the 6 letters. 82 1. Counting A piece of notation is helpful here: n!, read “n factorial”, is the product of all positive integers less than or equal to n (for reasons of convenience, we also deﬁne 0! to be 1). So the number of permutation of 6 letters, as seen in the previous example is 6! 6 · 5 · 4 · 3 · 2 · 1. This generalizes: Permutations of n elements. There are n! n · (n − 1) · (n − 2) · · · · · 2 · 1 permutations of n (distinct) elements. Example 1.3.2 Counting Bijective Functions. How many functions f : {1, 2, . . . , 8} → {1, 2, . . . , 8} are bĳective? Solution. Remember what it means for a function to be bĳective: each element in the codomain must be the image of exactly one element of the domain. Using two-line notation, we could write one of these bĳections as 1 2 3 4 5 6 7 8 . f 3 1 5 8 7 6 2 4 What we are really doing is just rearranging the elements of the codomain, so we are creating a permutation of 8 elements. In fact, “permutation” is another term used to describe bĳective functions from a ﬁnite set to itself. If you believe this, then you see the answer must be 8! 8 · 7 · · · · · 1 40320. You can see this directly as well: for each element of the domain, we must pick a distinct element of the codomain to map to. There are 8 choices for where to send 1, then 7 choices for where to send 2, and so on. We multiply using the multiplicative principle. Sometimes we do not want to permute all of the letters/numbers/ elements we are given. Example 1.3.3 How many 4 letter “words” can you make from the letters a through f, with no repeated letters? Solution. This is just like the problem of permuting 4 letters, only now we have more choices for each letter. For the ﬁrst letter, there are 6 choices. For each of those, there are 5 choices for the second letter. Then there are 4 choices for the third letter, and 3 choices for the last letter. The total number of words is 6 · 5 · 4 · 3 360. This is 1.3. Combinations and Permutations 83 not 6! because we never multiplied by 2 and 1. We could start with 6! and then cancel the 2 and 1, and thus write 6! 2! . In general, we can ask how many permutations exist of k objects choosing those objects from a larger collection of n objects. (In the example above, k 4, and n 6.) We write this number P(n, k) and sometimes call it a k-permutation of n elements. From the example above, we see that to compute P(n, k) we must apply the multiplicative principle to k numbers, starting with n and counting backwards. For example P(10, 4) 10 · 9 · 8 · 7. Notice again that P(10, 4) starts out looking like 10!, but we stop after 7. We can formally account for this “stopping” by dividing away the part of the factorial we do not want: P(10, 4) 10 · 9 · 8 · 7 · 6 · 5 · 4 · 3 · 2 · 1 10! . 6·5·4·3·2·1 6! Careful: The factorial in the denominator is not 4! but rather (10 − 4)!. k-permutations of n elements. P(n, k) is the number of k-permutations of n elements, the number of ways to arrange k objects chosen from n distinct objects. P(n, k) n! n(n − 1)(n − 2) · · · (n − (k − 1)). (n − k)! n! Note that when n k, we have P(n, n) (n−n)! n! (since we deﬁned 0! to be 1). This makes sense —we already know n! gives the number of permutations of all n objects. Example 1.3.4 Counting injective functions. How many functions f : {1, 2, 3} → {1, 2, 3, 4, 5, 6, 7, 8} are injective? Solution. Note that it doesn’t make sense to ask for the number of bĳections here, as there are none (because the codomain is larger than the domain, there are no surjections). But for a function to be injective, we just can’t use an element of the codomain more than once. We need to pick an element from the codomain to be the image of 1. There are 8 choices. Then we need to pick one of the remaining 7 elements to be the image of 2. Finally, one of the remaining 6 84 1. Counting elements must be the image of 3. So the total number of functions is 8 · 7 · 6 P(8, 3). What this demonstrates in general is that the number of injections f : A → B, where |A| k and |B| n, is P(n, k). Here is another way to ﬁnd the number of k-permutations of n elements: ﬁrst select which k elements will be in the permutation, then count how many ways there are to arrange them. Once you have selected the k objects, we know there are k! ways to arrange (permute) them. But how do you select k objects from the n? You have n objects, and you n need to choose k of them. You can do that in k ways. Then for each choice of those k elements, we can permute them in k! ways. Using the multiplicative principle, we get another formula for P(n, k): P(n, k) n · k!. k Now since we have a closed formula for P(n, k) already, we can substitute that in: n n! · k!. k (n − k)! If we divide both sides by k! we get a closed formula for Closed formula for n k . n k . n(n − 1)(n − 2) · · · (n − (k − 1)) n n! . k (n − k)!k! k(k − 1)(k − 2) · · · 1 We say P(n, k) counts permutations, and nk counts combinations. The formulas for each are very similar, there is just an extra k! in the denominator of nk . That extra k! accounts for the fact that nk does not distinguish between the diﬀerent orders that the k objects can appear in. We are just selecting (or choosing) the k objects, not arranging them. Perhaps “combination” is a misleading label. We don’t mean it like a combination lock (where the order would deﬁnitely matter). Perhaps a better metaphor is a combination of ﬂavors — you just need to decide which ﬂavors to combine, not the order in which to combine them. To further illustrate the connection between combinations and permutations, we close with an example. 1.3. Combinations and Permutations Example 1.3.5 You decide to have a dinner party. Even though you are incredibly popular and have 14 diﬀerent friends, you only have enough chairs to invite 6 of them. 1. How many choices do you have for which 6 friends to invite? 2. What if you need to decide not only which friends to invite but also where to seat them along your long table? How many choices do you have then? Solution. 1. You must simply choose 6 friends from a group of 14. This 14 can be done in 6 ways. We can ﬁnd this number either by 14! 3003. using Pascal’s triangle or the closed formula: 8!·6! 2. Here you must count all the ways you can permute 6 friends chosen from a group of 14. So the answer is P(14, 6), which can be calculated as 14! 8! 2162160. Notice that we can think of this counting problem as a question about counting functions: how many injective functions are there from your set of 6 chairs to your set of 14 friends (the functions are injective because you can’t have a single chair go to two of your friends). How are these numbers related? Notice that P(14, 6) is much larger than 14 . This makes sense. 14 6 6 picks 6 friends, but P(14, 6) arranges the 6 friends as well as picks them. In fact, we can say exactly how much larger P(14, 6) is. In both counting problems we choose 6 out of 14 friends. For the ﬁrst one, we stop there, at 3003 ways. But for the second counting problem, each of those 3003 choices of 6 friends can be arranged in exactly 6! ways. So now we have 3003 · 6! choices and that is exactly 2162160. Alternatively, look at the ﬁrst problem another way. We want to select 6 out of 14 friends, but we do not care about the order they are selected in. To select 6 out of 14 friends, we might try this: 14 · 13 · 12 · 11 · 10 · 9. This is a reasonable guess, since we have 14 choices for the ﬁrst guest, then 13 for the second, and so on. But the guess is wrong (in fact, that product is exactly 2162160 P(14, 6)). It distinguishes 85 86 1. Counting between the diﬀerent orders in which we could invite the guests. To correct for this, we could divide by the number of diﬀerent arrangements of the 6 guests (so that all of these would count as just one outcome). There are precisely 6! ways to arrange 6 guests, so the correct answer to the ﬁrst question is 14 · 13 · 12 · 11 · 10 · 9 . 6! Note that another way to write this is 14! . 8! · 6! which is what we had originally. Exercises 1. A pizza parlor oﬀers 10 toppings. (a) How many 3-topping pizzas could they put on their menu? Assume double toppings are not allowed. (b) How many total pizzas are possible, with between zero and ten toppings (but not double toppings) allowed? (c) The pizza parlor will list the 10 toppings in two equal-sized columns on their menu. How many ways can they arrange the toppings in the left column? 2. A combination lock consists of a dial with 40 numbers on it. To open the lock, you turn the dial to the right until you reach a ﬁrst number, then to the left until you get to second number, then to the right again to the third number. The numbers must be distinct. How many diﬀerent combinations are possible? 3. Using the digits 2 through 8, ﬁnd the number of diﬀerent 5-digit numbers such that: (a) Digits can be used more than once. (b) Digits cannot be repeated, but can come in any order. (c) Digits cannot be repeated and must be written in increasing order. (d) Which of the above counting questions is a combination and which is a permutation? Explain why this makes sense. 1.3. Combinations and Permutations 4. 87 In an attempt to clean up your room, you have purchased a new ﬂoating shelf to put some of your 17 books you have stacked in a corner. These books are all by diﬀerent authors. The new book shelf is large enough to hold 10 of the books. (a) How many ways can you select and arrange 10 of the 17 books on the shelf? Notice that here we will allow the books to end up in any order. Explain. (b) How many ways can you arrange 10 of the 17 books on the shelf if you insist they must be arranged alphabetically by author? Explain. 5. Suppose you wanted to draw a quadrilateral using the dots below as vertices (corners). The dots are spaced one unit apart horizontally and two units apart vertically. How many quadrilaterals are possible? How many are squares? How many are rectangles? How many are parallelograms? How many are trapezoids? (Here, as in calculus, a trapezoid is deﬁned as a quadrilateral with at least one pair of parallel sides. In particular, parallelograms are trapezoids.) How many are trapezoids that are not parallelograms? 6. How many triangles are there with vertices from the points shown below? Note, we are not allowing degenerate triangles - ones with all three vertices on the same line, but we do allow non-right triangles. Explain why your answer is correct. 88 1. Counting 7. An anagram of a word is just a rearrangement of its letters. How many diﬀerent anagrams of “uncopyrightable” are there? (This happens to be the longest common English word without any repeated letters.) 8. How many anagrams are there of the word “assesses” that start with the letter “a”? 9. How many anagrams are there of “anagram”? 10. On a business retreat, your company of 20 executives go golﬁng. (a) You need to divide up into foursomes (groups of 4 people): a ﬁrst foursome, a second foursome, and so on. How many ways can you do this? (b) After all your hard work, you realize that in fact, you want each foursome to include one of the ﬁve Board members. How many ways can you do this? 11. How many diﬀerent seating arrangements are possible for King Arthur and his 9 knights around their round table? 12. Consider sets A and B with |A| 10 and |B| 17. (a) How many functions f : A → B are there? (b) How many functions f : A → B are injective? 13. Consider functions f : {1, 2, 3, 4} → {1, 2, 3, 4, 5, 6}. (a) How many functions are there total? (b) How many functions are injective? (c) How many of the injective functions are increasing? To be increasing means that if a < b then f (a) < f (b), or in other words, the outputs get larger as the inputs get larger. 14. We have seen that the formula for P(n, k) is is to explain why this is the right formula. n! . Your task here (n − k)! (a) Suppose you have 12 chips, each a diﬀerent color. How many diﬀerent stacks of 5 chips can you make? Explain your answer and why it is the same as using the formula for P(12, 5). (b) Using the scenario of the 12 chips again, what does 12! count? What does 7! count? Explain. (c) Explain why it makes sense to divide 12! by 7! when computing P(12, 5) (in terms of the chips). (d) Does your explanation work for numbers other than 12 and 5? n! Explain the formula P(n, k) (n−k)! using the variables n and k. 1.4. Combinatorial Proofs 1.4 89 Combinatorial Proofs Investigate! 1. The Stanley Cup is decided in a best of 7 tournament between two teams. In how many ways can your team win? Let’s answer this question two ways: (a) How many of the 7 games does your team need to win? How many ways can this happen? (b) What if the tournament goes all 7 games? So you win the last game. How many ways can the ﬁrst 6 games go down? (c) What if the tournament goes just 6 games? How many ways can this happen? What about 5 games? 4 games? (d) What are the two diﬀerent ways to compute the number of ways your team can win? Write down an equation involving binomial coeﬃcients (that is, nk ’s). What pattern in Pascal’s triangle is this an example of? 2. Generalize. What if the rules changed and you played a best of 9 tournament (5 wins required)? What if you played an n game tournament with k wins required to be named champion? ! Attempt the above activity before proceeding ! Patterns in Pascal’s Triangle Have a look again at Pascal’s triangle. Forget for a moment where it comes from. Just look at it as a mathematical object. What do you notice? 1 1 2 1 3 1 1 6 1 1 7 3 10 35 1 4 1 5 10 20 15 21 1 6 4 5 1 1 6 15 35 1 21 1 7 1 90 1. Counting There are lots of patterns hidden away in the triangle, enough to ﬁll a reasonably sized book. Here are just a few of the most obvious ones: 1. The entries on the border of the triangle are all 1. 2. Any entry not on the border is the sum of the two entries above it. 3. The triangle is symmetric. In any row, entries on the left side are mirrored on the right side. 4. The sum of all entries on a given row is a power of 2. (You should check this!) We would like to state these observations in a more precise way, and then prove that they are correct. Now each entry in Pascal’s triangle is in fact a binomial coeﬃcient. The 1 on the very top of the triangle is 00 . The next row (which we will call row 1, even though it is not the top-most row) consists of 10 and 11 . Row 4 (the row 1, 4, 6, 4, 1) consists of the binomial coeﬃcients 4 4 4 4 4 . 0 1 2 3 4 Given this description of the elements in Pascal’s triangle, we can rewrite the above observations as follows: 1. 2. 3. 4. n 0 n k n k n 0 1 and + n−1 n n 1. n−1 k−1 + k . n n−k . n n 1 + 2 +··· + n n 2n . Each of these is an example of a binomial identity : an identity (i.e., equation) involving binomial coeﬃcients. Our goal is to establish these identities. We wish to prove that they hold for all values of n and k. These proofs can be done in many ways. One option would be to give algebraic proofs, using the formula for nk : n n! . k (n − k)! k! Here’s how you might do that for the second identity above. Example 1.4.1 Give an algebraic proof for the binomial identity n n−1 n−1 + . k k−1 k 91 1.4. Combinatorial Proofs Solution. Proof. By the deﬁnition of n k , we have (n − 1)! (n − 1)! n−1 k−1 (n − 1 − (k − 1))!(k − 1)! (n − k)!(k − 1)! and (n − 1)! n−1 . k (n − 1 − k)!k! Thus, starting with the right-hand side of the equation: (n − 1)! (n − 1)! n−1 n−1 + + k−1 k (n − k)!(k − 1)! (n − 1 − k)! k! (n − 1)!k (n − 1)!(n − k) + (n − k)! k! (n − k)! k! (n − 1)!(k + n − k) (n − k)! k! n! (n − k)! k! n . k The second line (where the common denominator is found) works because k(k − 1)! k! and (n − k)(n − k − 1)! (n − k)!. qed This is certainly a valid proof, but also is entirely useless. Even if you understand the proof perfectly, it does not tell you why the identity is true. A better approach would be to explain what nk means and then say why n−1 that is also what n−1 means. Let’s see how this works for the four k−1 + k identities we observed above. Example 1.4.2 Explain why n 0 1 and n n 1. Solution. What do these binomial coeﬃcients tell us? Well, n0 gives the number of ways to select 0 objects from a collection of n objects. There is only one way to do this, namely to not select any n n of the objects. Thus 0 1. Similarly, n gives the number of ways to select n objects from a collection of n objects. There is only n one way to do this: select all n objects. Thus n 1. Alternatively, we know that n0 is the number of n-bit strings with weight 0. There is only one such string, the string of all 0’s. 92 1. Counting So n0 1. Similarly nn is the number of n-bit strings with weight n. There is only one string with this property, the string of all 1’s. Another way: n0 gives the number of subsets of a set of size n containing 0 elements. There is only one such subset, the empty n set. n gives the number of subsets containing n elements. The only such subset is the original set (of all elements). Example 1.4.3 Explain why n k n−1 k−1 + n−1 k . Solution. The easiest way to see this is to consider bit strings. nk is the number of bit strings of length n containing k 1’s. Of all of these strings, some start with a 1 and the rest start with a 0. First consider all the bit strings which start with a 1. After the 1, there must be n − 1 more bits (to get the total length up to n) and exactly k − 1 of them must be 1’s (as we already have one, and we need k total). How many strings are there like that? There are exactly n−1 such bit strings, so of all the length n bit strings containing k−1 n−1 k 1’s, k−1 of them start with a 1. Similarly, there are n−1 which k start with a 0 (we still need n − 1 bits and now k of them must be n−1 1’s). Since there are k bit strings containing n − 1 bits with k 1’s, that is the number of length bitstrings with k 1’s which start n n−1 with a 0. Therefore nk n−1 + k−1 k . Another way: consider the question, how many ways can you select k pizza toppings from a menu containing n choices? One way to do this is just nk . Another way to answer the same question is to ﬁrst decide whether or not you want anchovies. If you do want anchovies, you still need to pick k − 1 toppings, now from just n − 1 choices. That can be done in n−1 k−1 ways. If you do not want anchovies, then you still need to select k toppings from n − 1 choices (the anchovies are out). You can do that in n−1 k ways. Since the choices with anchovies are disjoint the choices without n−1from anchovies, the total choices are n−1 + . But wait. We answered k−1 k the same question intwo diﬀerent ways, so the two answers must n−1 be the same. Thus nk n−1 + k−1 k . You can also explain (prove) this identity by counting subsets, or even lattice paths. 1.4. Combinatorial Proofs Example 1.4.4 Prove the binomial identity n k n n−k . Solution. Why is this true? nk counts the number of ways to n select k things from n choices. On the other hand, n−k counts the number of ways to select n − k things from n choices. Are these really the same? Well, what if instead of selecting the n − k things you choose to exclude them. How many ways are there to choose n n − k things to exclude from n choices. Clearly this is n−k as well (it doesn’t matter whether you include or exclude the things once you have chosen them). And if you exclude n − k things, then you are including the other k things. So the set of outcomes should be the same. Let’s try the pizza counting example like we did above. How many ways are there to pick k toppings from a list of n choices? On the one hand, the answer is simply nk . Alternatively, you could make a list of all the toppings you don’t want. To end up with a pizza containing exactly k toppings, you need to pick n −k toppings n to not put on the pizza. You have n−k choices for the toppings you don’t want. Both of these ways give you a pizza with k toppings, in fact all the ways to get a pizza k toppings. Thus these two with n . answers must be the same: nk n−k You can also prove (explain) this identity using bit strings, sub sets, or lattice paths. The bit string argument is nice: nk counts the number of bit strings of length n with k 1’s. This is also the number of bit string of length n with k 0’s (just replace each 1 with a 0 and each 0 with a 1). But if a string of length n has k 0’s, it must have n n − k 1’s. And there are exactly n−k strings of length n with n − k 1’s. Example 1.4.5 Prove the binomial identity n 0 + n 1 + n 2 +···+ n n 2n . Solution. Let’s do a “pizza proof” again. We need to ﬁnd a question about pizza toppings which has 2n as the answer. How about this: If a pizza joint oﬀers n toppings, how many pizzas can you build using any number of toppings from no toppings to all toppings, using each topping at most once? On one hand, the answer is 2n . For each topping you can say “yes” or “no,” so you have two choices for each topping. On the other hand, divide the possible pizzas into disjoint groups: the pizzas with no toppings, the pizzas with one top- 93 94 1. Counting ping, the pizzas with two toppings, etc. If we want no toppings, there is only one pizza like that (the empty pizza, if you will) but n it would be better to think of that number as 0 since we choose 0 of the n toppings. How many pizzas have 1 topping? We need to n choose 1 of the n toppings, so 1 . We have: • Pizzas with 0 toppings: • Pizzas with 1 topping: • Pizzas with 2 toppings: . • .. • Pizzas with n toppings: n 0 n 1 n 2 n n . The total number of possible pizzas will be the sum of these, which is exactly the left-hand side of the identity we are trying to prove. Again, we could have proved the identity using subsets, bit strings, or lattice paths (although the lattice path argument is a little tricky). Hopefully this gives some idea of how explanatory proofs of binomial identities can go. It is worth pointing out that more traditional proofs can also be beautiful.2 For example, consider the following rather slick proof of the last identity. Expand the binomial (x + y)n : n n n n−1 n n−2 2 n n n x + x y+ x y +···+ x · yn + y . (x + y) 0 1 2 n−1 n n Let x 1 and y 1. We get: n n n n−1 n n−2 2 n n n 1 + 1 1+ 1 1 +···+ 1 · 1n + 1 . (1 + 1) 0 1 2 n−1 n n Of course this simpliﬁes to: n n n n n (2) + + +···+ + . 0 1 2 n−1 n n Something fun to try: Let x 1 and y 2. Neat huh? 2Most every binomial identity can be proved using mathematical induction, using the recursive deﬁnition for nk . We will discuss induction in Section 2.5. 1.4. Combinatorial Proofs 95 More Proofs The explanatory proofs given in the above examples are typically called combinatorial proofs. In general, to give a combinatorial proof for a binomial identity, say A B you do the following: 1. Find a counting problem you will be able to answer in two ways. 2. Explain why one answer to the counting problem is A. 3. Explain why the other answer to the counting problem is B. Since both A and B are the answers to the same question, we must have A B. The tricky thing is coming up with the question. This is not always obvious, but it gets easier the more counting problems you solve. You will start to recognize types of answers as the answers to types of questions. More often what will happen is you will be solving a counting problem and happen to think up two diﬀerent ways of ﬁnding the answer. Now you have a binomial identity and the proof is right there. The proof is the problem you just solved together with your two solutions. For example, consider this counting question: How many 10-letter words use exactly four A’s, three B’s, two C’s and one D? Let’s try to solve this problem. We have 10 spots for letters to go. Four of those need to be A’s. We can pick the four A-spots in 10 4 ways. Now where can we put the B’s? Well there are only 6 spots left, we need to pick 6 3 of them. This can be done in 3 ways. The two C’s need to go in two of the 3 remaining spots, so we have 32 ways of doing that. That leaves just one spot of the D, but we could write that 1 choice as 11 . Thus the answer is: 10 6 3 1 . 4 3 2 1 But why stop there? We can ﬁnd the answer another way too. First let’s decide where to put the one D: we have 10 spots, we need to choose 1 10 9 of them, so this can be done in 1 ways. Next, choose one of the 2 ways to place the two C’s. We now have 7 spots left, and three of them need to be ﬁlled with B’s. There are 73 ways to do this. Finally the A’s can be placed in 44 (that is, only one) ways. So another answer to the question is 10 9 7 4 . 1 2 3 4 96 1. Counting Interesting. This gives us the binomial identity: 10 6 3 1 10 9 7 4 . 4 3 2 1 1 2 3 4 Here are a couple more binomial identities with combinatorial proofs. Example 1.4.6 Prove the identity n+2 1n + 2(n − 1) + 3(n − 2) + · · · + (n − 1)2 + n1 . 3 Solution. To give a combinatorial proof we need to think up a question we can answer in two ways: one way needs to give the left-hand-side of the identity, the other way needs to be the righthand-side of the identity. Our clue to what question to ask comes from the right-hand side: n+2 3 counts the number of ways to select 3 things from a group of n + 2 things. Let’s name those things 1, 2, 3, . . . , n + 2. In other words, we want to ﬁnd 3-element subsets of those numbers (since order should not matter, subsets are exactly the right thing to think about). We will have to be a bit clever to explain why the left-hand-side also gives the number of these subsets. Here’s the proof. Proof. Consider the question “How many 3-element subsets are there of the set {1, 2, 3, . . . , n + 2}?” We answer this in two ways: Answer 1: We must select 3 elements from the collection of n +2 n+2 elements. This can be done in 3 ways. Answer 2: Break this problem up into cases by what the middle number in the subset is. Say each subset is {a, b, c} written in increasing order. We count the number of subsets for each distinct value of b. The smallest possible value of b is 2, and the largest is n + 1. When b 2, there are 1 · n subsets: 1 choice for a and n choices (3 through n + 2) for c. When b 3, there are 2 · (n − 1) subsets: 2 choices for a and n − 1 choices for c. When b 4, there are 3 · (n − 2) subsets: 3 choices for a and n − 2 choices for c. And so on. When b n + 1, there are n choices for a and only 1 choice for c, so n · 1 subsets. Therefore the total number of subsets is 1n + 2(n − 1) + 3(n − 2) + · · · + (n − 1)2 + n1. 97 1.4. Combinatorial Proofs Since Answer 1 and Answer 2 are answers to the same question, they must be equal. Therefore n+2 . 1n + 2(n − 1) + 3(n − 2) + · · · + (n − 1)2 + n1 3 qed Example 1.4.7 Prove the binomial identity 2 n 0 2 n + 1 2 n + 2 2 n +···+ n 2n . n Solution. We will give two diﬀerent proofs of this fact. The ﬁrst will be very similar to the previous example (counting subsets). The second proof is a little slicker, using lattice paths. Proof. Consider the question: “How many pizzas can you make using n toppings when there are 2n toppings to choose from?” Answer 1: There are 2n toppings, from which you must choose n. This can be done in 2n n ways. Answer 2: Divide the toppings into two groups of n toppings (perhaps n meats and n veggies). Any choice of n toppings must include some number from the ﬁrst group and some number from the second group. Consider each possible number of meat toppings separately: 0 meats: n0 nn , since you need to choose 0 of the n meats and n of the n veggies. n , since you need 1 of n meats so n − 1 of n 1 meat: n1 n−1 veggies. n . Choose 2 meats and the remaining n − 2 2 meats: n2 n−2 toppings from the n veggies. And so on. The last case is n meats, which can be done in nn n0 ways. Thus the total number of pizzas possible is n 0 n n + n 1 n n + n−1 2 n n +···+ n−2 n n . 0 This is not quite the left-hand side . . . yet. Notice that nn n0 n and n−1 n1 and so on, by the identity in Example 1.4.4. Thus 98 1. Counting we do indeed get 2 n 0 2 n + 1 2 n + 2 2 +···+ n . n Since these two answers are answers to the same question, they must be equal, and thus 2 n 0 2 n + 1 2 n + 2 2 n +···+ n 2n . n qed For an alternative proof, we use lattice paths. This is reasonable to consider because the right-hand side of the identity reminds us of the number of paths from (0, 0) to (n, n). Proof. Consider the question: How many lattice paths are there from (0, 0) to (n, n)? Answer 1: We must travel 2n steps, and n of them must be in 2n the up direction. Thus there are n paths. Answer 2: Note that any path from (0, 0) to (n, n) must cross the line x + y n. That is, any path must pass through exactly one of the points: (0, n), (1, n − 1), (2, n − 2), . . . , (n, 0). For example, this is what happens in the case n 4: (0,4) (4,4) (1,3) (2,2) (3,1) (4,0) (0,0) x+y4 How many paths pass through (0, n)? To get to that point, you must travel n units, and 0 of them are to the right, so there are n ways to get to (0, n). From (0, n) to (n, n) takes n steps, and 0 0 n of them are up. So there are 0 ways to get from (0, n) to (n, n). Therefore there are n0 n0 paths from (0, 0) to (n, n) through the point (0, n). What about through (1, n −1). There are n1 paths to get there (n steps, 1 to the right) and n1 paths to complete the journey to (n, n) (n steps, 1 up). So there are n1 n1 paths from (0, 0) to (n, n) through (1, n − 1). In general, to get to (n, n) through the point (k, n − k) we have n n k paths to the midpoint and then k paths from the midpoint 1.4. Combinatorial Proofs 99 to (n, n). So there are nk nk paths from (0, 0) to (n, n) through (k, n − k). All together then the total paths from (0, 0) to (n, n) passing through exactly one of these midpoints is 2 n 0 2 n + 1 2 n + 2 2 +···+ n . n Since these two answers are answers to the same question, they must be equal, and thus 2 n 0 2 n + 1 2 n + 2 2 n +···+ n 2n . n qed Exercises 1. Give a combinatorial proof of the identity 2 + 2 + 2 3 · 2. 2. Suppose you own x fezzes and y bow ties. Of course, x and y are both greater than 1. (a) How many combinations of fez and bow tie can you make? You can wear only one fez and one bow tie at a time. Explain. (b) Explain why the answer is also x+y − x2 − 2y . (If this is what 2 you claimed the answer was in part (a), try it again.) (c) Use your answers to parts (a) and (b) to give a combinatorial proof of the identity 3. x+y y x − − x y.. 2 2 2 How many triangles can you draw using the dots below as vertices? (a) Find an expression for the answer which is the sum of three terms involving binomial coeﬃcients. (b) Find an expression for the answer which is the diﬀerence of two binomial coeﬃcients. 100 1. Counting (c) Generalize the above to state and prove a binomial identity using a combinatorial proof. Say you have x points on the horizontal axis and y points in the semi-circle. 4. Consider all the triangles you can create using the points shown below as vertices. Note, we are not allowing degenerate triangles (ones with all three vertices on the same line) but we do allow non-right triangles. (a) Find the number of triangles, and explain why your answer is correct. (b) Find the number of triangles again, using a diﬀerent method. Explain why your new method works. (c) State a binomial identity that your two answers above establish (that is, give the binomial identity that your two answers a proof for). Then generalize this using m’s and n’s. 5. A woman is getting married. She has 15 best friends but can only select 6 of them to be her bridesmaids, one of which needs to be her maid of honor. How many ways can she do this? (a) What if she ﬁrst selects the 6 bridesmaids, and then selects one of them to be the maid of honor? (b) What if she ﬁrst selects her maid of honor, and then 5 other bridesmaids? (c) Explain why 6 6. 15 6 15 14 5 . Consider the identity: n n−1 k n . k k−1 (a) Is this true? Try it for a few values of n and k. (b) Use the formula for n k to give an algebraic proof of the identity. (c) Give a combinatorial proof of the identity. 1.4. Combinatorial Proofs 7. Give a combinatorial proof of the identity 8. Consider the binomial identity n n−2 2 k−2 101 n k k 2 . n n n n +2 +3 +···+ n n2n−1 . 1 2 3 n (a) Give a combinatorial proof of this identity. Hint: What if some number of a group of n people wanted to go to an escape room, and among those going, one needed to be the team captain? (b) Give an alternate proof by multiplying out (1 + x)n and taking derivatives of both sides. 9. Give a combinatorial proof for the identity 1 + 2 + 3 + · · · + n n+1 2 . 10. Consider the bit strings in B62 (bit strings of length 6 and weight 2). (a) How many of those bit strings start with 1? (b) How many of those bit strings start with 01? (c) How many of those bit strings start with 001? (d) Are there any other strings we have not counted yet? Which ones, and how many are there? (e) How many bit strings are there total in B62 ? (f) What binomial identity have you just given a combinatorial proof for? 11. Let’s count ternary digit strings, that is, strings in which each digit can be 0, 1, or 2. (a) How many ternary digit strings contain exactly n digits? (b) How many ternary digit strings contain exactly n digits and n 2’s. (c) How many ternary digit strings contain exactly n digits and n − 1 2’s. (Hint: where can you put the non-2 digit, and then what could it be?) (d) How many ternary digit strings contain exactly n digits and n − 2 2’s. (Hint: see previous hint) (e) How many ternary digit strings contain exactly n digits and n − k 2’s. (f) How many ternary digit strings contain exactly n digits and no 2’s. (Hint: what kind of a string is this?) 102 1. Counting (g) Use the above parts to give a combinatorial proof for the identity n n n n n +2 + 22 + 23 + · · · + 2n 3n . 0 1 2 3 n 12. How many ways are there to rearrange the letters in the word “rearrange”? Answer this question in at least two diﬀerent ways to establish a binomial identity. 13. Establish the identity below using a combinatorial proof. 2 2 n 3 + 2 2 n−1 4 + 2 2 n−2 n +···+ 2 2 2 n+3 . 2 5 14. In Example 1.4.5 we established that the sum of any row in Pascal’s triangle is a power of two. Speciﬁcally, n n n n + + +···+ 2n . 0 1 2 n The argument given there used the counting question, “how many pizzas can you build using any number of n diﬀerent toppings?” To practice, give new proofs of this identity using diﬀerent questions. (a) Use a question about counting subsets. (b) Use a question about counting bit strings. (c) Use a question about counting lattice paths. 1.5. Stars and Bars 1.5 103 Stars and Bars Investigate! Suppose you have some number of identical Rubik’s cubes to distribute to your friends. Imagine you start with a single row of the cubes. 1. Find the number of diﬀerent ways you can distribute the cubes provided: (a) You have 3 cubes to give to 2 people. (b) You have 4 cubes to give to 2 people. (c) You have 5 cubes to give to 2 people. (d) You have 3 cubes to give to 3 people. (e) You have 4 cubes to give to 3 people. (f) You have 5 cubes to give to 3 people. 2. Make a conjecture about how many diﬀerent ways you could distribute 7 cubes to 4 people. Explain. 3. What if each person were required to get at least one cube? How would your answers change? ! Attempt the above activity before proceeding ! Consider the following counting problem: You have 7 cookies to give to 4 kids. How many ways can you do this? Take a moment to think about how you might solve this problem. You may assume that it is acceptable to give a kid no cookies. Also, the cookies are all identical and the order in which you give out the cookies does not matter. Before solving the problem, here is a wrong answer: You might guess that the answer should be 47 because for each of the 7 cookies, there are 4 choices of kids to which you can give the cookie. This is reasonable, but wrong. To see why, consider a few possible outcomes: we could assign the ﬁrst six cookies to kid A, and the seventh cookie to kid B. Another outcome would assign the ﬁrst cookie to kid B and the six remaining cookies to kid A. Both outcomes are included in the 47 answer. But for 104 1. Counting our counting problem, both outcomes are really the same – kid A gets six cookies and kid B gets one cookie. What do outcomes actually look like? How can we represent them? One approach would be to write an outcome as a string of four numbers like this: 3112, which represent the outcome in which the ﬁrst kid gets 3 cookies, the second and third kid each get 1 cookie, and the fourth kid gets 2 cookies. Represented this way, the order in which the numbers occur matters. 1312 is a diﬀerent outcome, because the ﬁrst kid gets a one cookie instead of 3. Each number in the string can be any integer between 0 and 7. But the answer is not 74 . We need the sum of the numbers to be 7. Another way we might represent outcomes is to write a string of seven letters: ABAADCD, which represents that the ﬁrst cookie goes to kid A, the second cookie goes to kid B, the third and fourth cookies go to kid A, and so on. In fact, this outcome is identical to the previous one—A gets 3 cookies, B and C get 1 each and D gets 2. Each of the seven letters in the string can be any of the 4 possible letters (one for each kid), but the number of such strings is not 47 , because here order does not matter. In fact, another way to write the same outcome is AAABCDD. This will be the preferred representation of the outcome. Since we can write the letters in any order, we might as well write them in alphabetical order for the purposes of counting. So we will write all the A’s ﬁrst, then all the B’s, and so on. Now think about how you could specify such an outcome. All we really need to do is say when to switch from one letter to the next. In terms of cookies, we need to say after how many cookies do we stop giving cookies to the ﬁrst kid and start giving cookies to the second kid. And then after how many do we switch to the third kid? And after how many do we switch to the fourth? So yet another way to represent an outcome is like this: ∗ ∗ ∗| ∗ | ∗ | ∗ ∗. Three cookies go to the ﬁrst kid, then we switch and give one cookie to the second kid, then switch, one to the third kid, switch, two to the fourth kid. Notice that we need 7 stars and 3 bars – one star for each cookie, and one bar for each switch between kids, so one fewer bars than there are kids (we don’t need to switch after the last kid – we are done). Why have we done all of this? Simple: to count the number of ways to distribute 7 cookies to 4 kids, all we need to do is count how many 1.5. Stars and Bars 105 stars and bars charts there are. But a stars and bars chart is just a string of symbols, some stars and some bars. If instead of stars and bars we would use 0’s and 1’s, it would just be a bit string. We know how to count those. Before we get too excited, we should make sure that really any string of (in our case) 7 stars and 3 bars corresponds to a diﬀerent way to distribute cookies to kids. In particular consider a string like this: | ∗ ∗ ∗ || ∗ ∗ ∗ ∗. Does that correspond to a cookie distribution? Yes. It represents the distribution in which kid A gets 0 cookies (because we switch to kid B before any stars), kid B gets three cookies (three stars before the next bar), kid C gets 0 cookies (no stars before the next bar) and kid D gets the remaining 4 cookies. No matter how the stars and bars are arranged, we can distribute cookies in that way. Also, given any way to distribute cookies, we can represent that with a stars and bars chart. For example, the distribution in which kid A gets 6 cookies and kid B gets 1 cookie has the following chart: ∗ ∗ ∗ ∗ ∗ ∗ | ∗ ||. After all that work we are ﬁnally ready to count. Each way to distribute cookies corresponds to a stars and bars chart with 7 stars and 3 bars. So there are 10 symbols, and we must choose 3 of them to be bars. Thus: There are 10 ways to distribute 7 cookies to 4 kids. 3 While we are at it, we can also answer a related question: how many ways are there to distribute 7 cookies to 4 kids so that each kid gets at least one cookie? What can you say about the corresponding stars and bars charts? The charts must start and end with at least one star (so that kids A and D) get cookies, and also no two bars can be adjacent (so that kids B and C are not skipped). One way to assure this is to place bars only in the spaces between the stars. With 7 stars, there are 6 spots between the stars, so we must choose 3 of those 6 spots to ﬁll with bars. Thus there are 63 ways to distribute 7 cookies to 4 kids giving at least one cookie to each kid. Another (and more general) way to approach this modiﬁed problem is to ﬁrst give each kid one cookie. Now the remaining 3 cookies can be distributed to the 4 kids without restrictions. So we have 3 stars and 3 bars for a total of 6 symbols, 3 of which must be bars. So again we see 6 that there are 3 ways to distribute the cookies. Stars and bars can be used in counting problems other than kids and cookies. Here are a few examples: 106 1. Counting Example 1.5.1 Your favorite mathematical ice-cream parlor oﬀers 10 ﬂavors. How many milkshakes could you create using exactly 6, not necessarily distinct scoops? The order you add the ﬂavors does not matter (they will be blended up anyway) but you are allowed repeats. So one possible shake is triple chocolate, double cherry, and mint chocolate chip. Solution. We get six scoops, each of which could be one of ten possible ﬂavors. Represent each scoop as a star. Think of going down the counter one ﬂavor at a time: you see vanilla ﬁrst, and skip to the next, chocolate. You say yes to chocolate three times (use three stars), then switch to the next ﬂavor. You keep skipping until you get to cherry, which you say yes to twice. Another switch and you are at mint chocolate chip. You say yes once. Then you keep switching until you get past the last ﬂavor, never saying yes again (since you already have said yes six times). There are ten ﬂavors to choose from, so we must switch from considering one ﬂavor to the next nine times. These are the nine bars. Now that we are conﬁdent that we have the right number of stars and bars, we answer the question simply: there are 6 stars and 9 bars, so 15 symbols. We need to pick 9 of them to be bars, so the number of milkshakes possible is 15 9 . Example 1.5.2 How many 7 digit phone numbers are there in which the digits are non-increasing? That is, every digit is less than or equal to the previous one. Solution. We need to decide on 7 digits so we will use 7 stars. The bars will represent a switch from each possible single digit number down to the next smaller one. So the phone number 866-5221 is represented by the stars and bars chart | ∗ || ∗ ∗| ∗ ||| ∗ ∗| ∗ |. There are 10 choices for each digit (0-9) so we must switch between choices 9 times. We have 7 stars and 9 bars, so the total number of phone numbers is 16 9 . 1.5. Stars and Bars 107 Example 1.5.3 How many integer solutions are there to the equation x 1 + x2 + x 3 + x 4 + x 5 13. (An integer solution to an equation is a solution in which the unknown must have an integer value.) 1. where x i ≥ 0 for each x i ? 2. where x i > 0 for each x i ? 3. where x i ≥ 2 for each x i ? Solution. This problem is just like giving 13 cookies to 5 kids. We need to say how many of the 13 units go to each of the 5 variables. In other words, we have 13 stars and 4 bars (the bars are like the “+” signs in the equation). 1. If x i can be 0 or greater, we are in the standard case with no restrictions. So 13 stars and 4 bars can be arranged in 17 4 ways. 2. Now each variable must be at least 1. So give one unit to each variable to satisfy that restriction. Now there are 8 stars left, and still 4 bars, so the number of solutions is 12 4 . 3. Now each variable must be 2 or greater. So before any counting, give each variable 2 units. We now have 3 remaining stars and 4 bars, so there are 74 solutions. Counting with Functions. Many of the counting problems in this section might at ﬁrst appear to be examples of counting functions. After all, when we try to count the number of ways to distribute cookies to kids, we are assigning each cookie to a kid, just like you assign elements of the domain of a function to elements in the codomain. However, the number of ways to assign 7 cookies to 4 108 1. Counting kids is 10 7 120, while the number of functions f : {1, 2, 3, 4, 5, 6, 7} → {a, b, c, d} is 47 16384. What is going on here? When we count functions, we consider the following two functions, for example, to be diﬀerent: 1 2 3 4 5 6 7 f a b c c c c c 1 2 3 4 5 6 7 g . b a c c c c c But these two functions would correspond to the same cookie distribution: kids a and b each get one cookie, kid c gets the rest (and none for kid d). The point: elements of the domain are distinguished, cookies are indistinguishable. This is analogous to the distinction between permutations (like counting functions) and combinations (not). Exercises 1. A multiset is a collection of objects, just like a set, but can contain an object more than once (the order of the elements still doesn’t matter). For example, {1, 1, 2, 5, 5, 7} is a multiset of size 6. (a) How many sets of size 5 can be made using the 10 numeric digits 0 through 9? (b) How many multisets of size 5 can be made using the 10 numeric digits 0 through 9? 2. Using the digits 2 through 8, ﬁnd the number of diﬀerent 5-digit numbers such that: (a) Digits cannot be repeated and must be written in increasing order. For example, 23678 is okay, but 32678 is not. (b) Digits can be repeated and must be written in non-decreasing order. For example, 24448 is okay, but 24484 is not. 3. Each of the counting problems below can be solved with stars and bars. For each, say what outcome the diagram ∗ ∗ ∗| ∗ || ∗ ∗| represents, if there are the correct number of stars and bars for the problem. Otherwise, say why the diagram does not represent any outcome, and what a correct diagram would look like. (a) How many ways are there to select a handful of 6 jellybeans from a jar that contains 5 diﬀerent ﬂavors? (b) How many ways can you distribute 5 identical lollipops to 6 kids? (c) How many 6-letter words can you make using the 5 vowels? 1.5. Stars and Bars 109 (d) How many solutions are there to the equation x 1 +x 2 +x 3 +x 4 6. 4. After gym class you are tasked with putting the 14 identical dodgeballs away into 5 bins. (a) How many ways can you do this if there are no restrictions? (b) How many ways can you do this if each bin must contain at least one dodgeball? 5. How many integer solutions are there to the equation x + y + z 8 for which (a) x, y, and z are all positive? (b) x, y, and z are all non-negative? (c) x, y, and z are all greater than or equal to −3. 6. When playing Yahtzee, you roll ﬁve regular 6-sided dice. How many diﬀerent outcomes are possible from a single roll? The order of the dice does not matter. 7. Your friend tells you she has 7 coins in her hand (just pennies, nickels, dimes and quarters). If you guess how many of each kind of coin she has, she will give them to you. If you guess randomly, what is the probability that you will be correct? 8. How many integer solutions to x 1 + x 2 + x 3 + x 4 25 are there for which x 1 ≥ 1, x2 ≥ 2, x 3 ≥ 3 and x 4 ≥ 4? 9. Solve the three counting problems below. Then say why it makes sense that they all have the same answer. That is, say how you can interpret them as each other. (a) How many ways are there to distribute 8 cookies to 3 kids? (b) How many solutions in non-negative integers are there to x + y + z 8? (c) How many diﬀerent packs of 8 crayons can you make using crayons that come in red, blue and yellow? 10. Consider functions f : {1, 2, 3, 4, 5} → {0, 1, 2, . . . , 9}. (a) How many of these functions are strictly increasing? Explain. (A function is strictly increasing provided if a < b, then f (a) < f (b).) (b) How many of the functions are non-decreasing? Explain. (A function is non-decreasing provided if a < b, then f (a) ≤ f (b).) 110 1. Counting 11. Conic, your favorite math themed fast food drive-in oﬀers 20 ﬂavors which can be added to your soda. You have enough money to buy a large soda with 4 added ﬂavors. How many diﬀerent soda concoctions can you order if: (a) You refuse to use any of the ﬂavors more than once? (b) You refuse repeats but care about the order the ﬂavors are added? (c) You allow yourself multiple shots of the same ﬂavor? (d) You allow yourself multiple shots, and care about the order the ﬂavors are added? 1.6. Advanced Counting Using PIE 1.6 111 Advanced Counting Using PIE Investigate! You have 11 identical mini key-lime pies to give to 4 children. However, you don’t want any kid to get more than 3 pies. How many ways can you distribute the pies? 1. How many ways are there to distribute the pies without any restriction? 2. Let’s get rid of the ways that one or more kid gets too many pies. How many ways are there to distribute the pies if Al gets too many pies? What if Bruce gets too many? Or Cat? Or Dent? 3. What if two kids get too many pies? How many ways can this happen? Does it matter which two kids you pick to overfeed? 4. Is it possible that three kids get too many pies? If so, how many ways can this happen? 5. How should you combine all the numbers you found above to answer the original question? Suppose now you have 13 pies and 7 children. No child can have more than 2 pies. How many ways can you distribute the pies? ! Attempt the above activity before proceeding ! Stars and bars allows us to count the number of ways to distribute 10 cookies to 3 kids and natural number solutions to x + y + z 11, for example. A relatively easy modiﬁcation allows us to put a lower bound restriction on these problems: perhaps each kid must get at least two cookies or x, y, z ≥ 2. This was done by ﬁrst assigning each kid (or variable) 2 cookies (or units) and then distributing the rest using stars and bars. What if we wanted an upper bound restriction? For example, we might insist that no kid gets more than 4 cookies or that x, y, z ≤ 4. It turns out this is considerably harder, but still possible. The idea is to count all the distributions and then remove those that violate the condition. In other words, we must count the number of ways to distribute 11 cookies to 3 kids in which one or more of the kids gets more than 4 cookies. For any particular kid, this is not a problem; we do this using stars and bars. But 112 1. Counting how to combine the number of ways for kid A, or B or C? We must use the PIE. The Principle of Inclusion/Exclusion (PIE) gives a method for ﬁnding the cardinality of the union of not necessarily disjoint sets. We saw in Section 1.1 how this works with three sets. To ﬁnd how many things are in one or more of the sets A, B, and C, we should just add up the number of things in each of these sets. However, if there is any overlap among the sets, those elements are counted multiple times. So we subtract the things in each intersection of a pair of sets. But doing this removes elements which are in all three sets once too often, so we need to add it back in. In terms of cardinality of sets, we have |A ∪ B ∪ C| |A| + |B| + |C| − |A ∩ B| − |A ∩ C| − |B ∩ C| + |A ∩ B ∩ C|. Example 1.6.1 Three kids, Alberto, Bernadette, and Carlos, decide to share 11 cookies. They wonder how many ways they could split the cookies up provided that none of them receive more than 4 cookies (someone receiving no cookies is for some reason acceptable to these kids). the “no more than 4” restriction, the answer Solution. Without 13 would be 2 , using 11 stars and 2 bars (separating the three kids). Now count the number of ways that one or more of the kids violates the condition, i.e., gets at least 4 cookies. Let A be the set of outcomes in which Alberto gets more than 4 cookies. Let B be the set of outcomes in which Bernadette gets more than 4 cookies. Let C be the set of outcomes in which Carlos gets more than 4 cookies. We then are looking (for the sake of subtraction) for the size of the set A ∪ B ∪ C. Using PIE, we must ﬁnd the sizes of |A|, |B|, |C|, |A ∩ B| and so on. Here is what we ﬁnd. • |A| 82 . First give Alberto 5 cookies, then distribute the remaining 6 to the three kids without restrictions, using 6 stars and 2 bars. • |B| 82 . Just like above, only now Bernadette gets 5 cookies at the start. • |C| 8 2 . Carlos gets 5 cookies ﬁrst. • |A ∩ B| 32 . Give Alberto and Bernadette 5 cookies each, leaving 1 (star) to distribute to the three kids (2 bars). • |A ∩ C| 3 2 . Alberto and Carlos get 5 cookies ﬁrst. 1.6. Advanced Counting Using PIE • |B ∩ C| 3 2 . 113 Bernadette and Carlos get 5 cookies ﬁrst. • |A ∩ B ∩ C| 0. It is not possible for all three kids to get 4 or more cookies. Combining all of these we see 8 8 8 3 3 3 |A ∪ B ∪ C| + + − − − + 0 75. 2 2 2 2 2 2 Thus the answer to the original question is 13 2 −75 78−75 3. This makes sense now that we see it. The only way to ensure that no kid gets more than 4 cookies is to give two kids 4 cookies and one kid 3; there are three choices for which kid that should be. We could have found the answer much quicker through this observation, but the point of the example is to illustrate that PIE works! For four or more sets, we do not write down a formula for PIE. Instead, we just think of the principle: add up all the elements in single sets, then subtract out things you counted twice (elements in the intersection of a pair of sets), then add back in elements you removed too often (elements in the intersection of groups of three sets), then take back out elements you added back in too often (elements in the intersection of groups of four sets), then add back in, take back out, add back in, etc. This would be very diﬃcult if it wasn’t for the fact that in these problems, all the cardinalities of the single sets are equal, as are all the cardinalities of the intersections of two sets, and that of three sets, and so on. Thus we can group all of these together and multiply by how many diﬀerent combinations of 1, 2, 3, . . . sets there are. Example 1.6.2 How many ways can you distribute 10 cookies to 4 kids so that no kid gets more than 2 cookies? Solution. There are 13 3 ways to distribute 10 cookies to 4 kids (using 10 stars and 3 bars). We will subtract all the outcomes in which a kid gets 3 or more cookies. How many outcomes are there like that? We can force kid A to eat 3 or more cookies by giving him 3 cookies before we start. Doing so reduces the problem to one in which we have 7 cookies to give to 4 kids without any restrictions. In that case, we have 7 stars (the 7 remaining cookies) and 3 bars (one less than the number of kids) so we can distribute the cookies in 10 3 ways. Of course we could choose any one of the 4 kids to 114 1. Counting give too many cookies, so it would appear that there are 41 10 3 ways to distribute the cookies giving too many to one kid. But in fact, we have over counted. We must get rid of the outcomes in which two kids have too many cookies. There are 42 ways to select 2 kids to give extra cookies. It takes 6 cookies to do this, leaving only 4 cookies. So we have 4 stars and still 3 bars. The remaining 4 cookies can thus be 7 4 distributed in 3 ways (for each of the 2 choices of which 2 kids to over-feed). But now we have removed too much. We must add back in all the ways to give too many cookies to three kids. This uses 9 cookies, leaving only 1 to distribute to the 4 kids using stars and bars, which can be done in 43 ways. We must consider this outcome for every possible choice of which three kids we over-feed, and there are 43 ways of selecting that set of 3 kids. Next we would subtract all the ways to give four kids too many cookies, but in this case, that number is 0. All together we get that the number of ways to distribute 10 cookies to 4 kids without giving any kid more than 2 cookies is: 13 − 3 4 10 4 7 4 4 − + 1 3 2 3 3 3 which is 286 − [480 − 210 + 16] 0. This makes sense: there is NO way to distribute 10 cookies to 4 kids and make sure that nobody gets more than 2. It is slightly surprising that 13 3 4 10 4 7 4 4 − + 1 3 2 3 3 3 , but since PIE works, this equality must hold. Just so you don’t think that these problems always have easier solutions, consider the following example. Example 1.6.3 Earlier (Example 1.5.3) we counted the number of solutions to the equation x 1 + x 2 + x 3 + x 4 + x 5 13, 1.6. Advanced Counting Using PIE 115 where x i ≥ 0 for each x i . How many of those solutions have 0 ≤ x i ≤ 3 for each x i ? Solution. We must subtract oﬀ the number of solutions in which one or more of the variables has a value greater than 3. We will need to use PIE because counting the number of solutions for which each of the ﬁve variables separately are greater than 3 counts solutions multiple times. Here is what we get: • Total solutions: 17 4 . • Solutions where x 1 > 3: 13 4 . Give x 1 4 units ﬁrst, then distribute the remaining 9 units to the 5 variables. • Solutions where x 1 > 3 and x 2 > 3: 94 . After you give 4 units to x 1 and another 4 to x2 , you only have 5 units left to distribute. • Solutions where x 1 > 3, x 2 > 3 and x 3 > 3: 5 4 . • Solutions where x 1 > 3, x 2 > 3, x 3 > 3, and x 4 > 3: 0. We also need to account for the fact that we could choose any of the ﬁve variables in the place of x 1 above (so there will be 51 outcomes like this), any pair of variables in the place of x 1 and x 2 ( 52 outcomes) and so on. It is because of this that the double counting occurs, so we need to use PIE. All together we have that the number of solutions with 0 ≤ x i ≤ 3 is 17 − 4 5 13 5 9 5 5 − + 1 4 2 4 3 4 15. Counting Derangements Investigate! For your senior prank, you decide to switch the nameplates on your favorite 5 professors’ doors. So that none of them feel left out, you want to make sure that all of the nameplates end up on the wrong door. How many ways can this be accomplished? ! Attempt the above activity before proceeding ! The advanced use of PIE has applications beyond stars and bars. A derangement of n elements {1, 2, 3, . . . , n} is a permutation in which no 116 1. Counting element is ﬁxed. For example, there are 6 permutations of the three elements {1, 2, 3}: 123 132 213 231 312 321. but most of these have one or more elements ﬁxed: 123 has all three elements ﬁxed since all three elements are in their original positions, 132 has the ﬁrst element ﬁxed (1 is in its original ﬁrst position), and so on. In fact, the only derangements of three elements are 231 and 312. If we go up to 4 elements, there are 24 permutations (because we have 4 choices for the ﬁrst element, 3 choices for the second, 2 choices for the third leaving only 1 choice for the last). How many of these are derangements? If you list out all 24 permutations and eliminate those which are not derangements, you will be left with just 9 derangements. Let’s see how we can get that number using PIE. Example 1.6.4 How many derangements are there of 4 elements? Solution. We count all permutations, and subtract those which are not derangements. There are 4! 24 permutations of 4 elements. Now for a permutation to not be a derangement, at least one of the 4 elements must be ﬁxed. There are 41 choices for which single element we ﬁx. Once ﬁxed, we need to ﬁnd a permutation of the other three elements. There are 3! permutations on 3 elements. But now we have counted too many non-derangements, so we must subtract those permutations which ﬁx two elements. There are 42 choices for which two elements we ﬁx, and then for each pair, 2! permutations of the remaining elements. But this subtracts too many, so add back in permutations which ﬁx 3 elements, all 4 4 1! of them. Finally subtract the 0! permutations (recall 0! 1) 3 4 which ﬁx all four elements. All together we get that the number of derangements of 4 elements is: 4! − 4 4 4 4 3! − 2! + 1! − 0! 24 − 15 9. 1 2 3 4 Of course we can use a similar formula to count the derangements of any number of elements. However, the more elements we have, the longer the formula gets. Here is another example: 1.6. Advanced Counting Using PIE 117 Example 1.6.5 Five gentlemen attend a party, leaving their hats at the door. At the end of the party, they hastily grab hats on their way out. How many diﬀerent ways could this happen so that none of the gentlemen leave with his own hat? Solution. We are counting derangements on 5 elements. There are 5! ways for the gentlemen to grab hats in any order—but many of these permutations will result in someone getting their own hat. So we subtract all the ways in which one or more of the men get their own hat. In other words, we subtract the non-derangements. Doing so requires PIE. Thus the answer is: 5! − 5 5 5 5 5 4! − 3! + 2! − 1! + 0! . 1 2 3 4 5 Counting Functions Investigate! 1. Consider all functions f : {1, 2, 3, 4, 5} → {1, 2, 3, 4, 5}. How many functions are there all together? How many of those are injective? Remember, a function is an injection if every input goes to a diﬀerent output. 2. Consider all functions f : {1, 2, 3, 4, 5} → {1, 2, 3, 4, 5}. How many of the injections have the property that f (x) , x for any x ∈ {1, 2, 3, 4, 5}? Your friend claims that the answer is: 5! − 5 5 5 5 5 4! − 3! + 2! − 1! + 0! . 1 2 3 4 5 Explain why this is correct. 3. Recall that a surjection is a function for which every element of the codomain is in the range. How many of the functions f : {1, 2, 3, 4, 5} → {1, 2, 3, 4, 5} are surjective? Use PIE! ! Attempt the above activity before proceeding ! We have seen throughout this chapter that many counting questions can be rephrased as questions about counting functions with certain properties. This is reasonable since many counting questions can be thought 118 1. Counting of as counting the number of ways to assign elements from one set to elements of another. Example 1.6.6 You decide to give away your video game collection so as to better spend your time studying advanced mathematics. How many ways can you do this, provided: 1. You want to distribute your 3 diﬀerent PS4 games among 5 friends, so that no friend gets more than one game? 2. You want to distribute your 8 diﬀerent 3DS games among 5 friends? 3. You want to distribute your 8 diﬀerent SNES games among 5 friends, so that each friend gets at least one game? In each case, model the counting question as a function counting question. Solution. 1. We must use the three games (call them 1, 2, 3) as the domain and the 5 friends (a,b,c,d,e) as the codomain (otherwise the function would not be deﬁned for the whole domain when a friend didn’t get any game). So how many functions are there with domain {1, 2, 3} and codomain {a, b, c, d, e}? The answer to this is 53 125, since we can assign any of 5 elements to be the image of 1, any of 5 elements to be the image of 2 and any of 5 elements to be the image of 3. But this is not the correct answer to our counting problem, 1 2 3 ; one friend because one of these functions is f a a a can get more than one game. What we really need to do is count injective functions. This gives P(5, 3) 60 functions, which is the answer to our counting question. 2. Again, we need to use the 8 games as the domain and the 5 friends as the codomain. We are counting all functions, so the number of ways to distribute the games is 58 . 3. This question is harder. Use the games as the domain and friends as the codomain (the reverse would not give a function). To ensure that every friend gets at least one game means that every element of the codomain is in the range. In other words, we are looking for surjective functions. How do you count those?? 1.6. Advanced Counting Using PIE 119 In Example 1.1.5 we saw how to count all functions (using the multiplicative principle) and in Example 1.3.4 we learned how to count injective functions (using permutations). Surjective functions are not as easily counted (unless the size of the domain is smaller than the codomain, in which case there are none). The idea is to count the functions which are not surjective, and then subtract that from the total number of functions. This works very well when the codomain has two elements in it: Example 1.6.7 How many functions f : {1, 2, 3, 4, 5} → {a, b} are surjective? Solution. There are 25 functions all together, two choices for where to send each of the 5 elements of the domain. Now of these, the functions which are not surjective must exclude one or more elements of the codomain from the range. So ﬁrst, consider functions for which a is not in the range. This can only happen one way: everything gets sent to b. Alternatively, we could exclude b from the range. Then everything gets sent to a, so there is only one function like this. These are the only ways in which a function could not be surjective (no function excludes both a and b from the range) so there are exactly 25 − 2 surjective functions. When there are three elements in the codomain, there are now three choices for a single element to exclude from the range. Additionally, we could pick pairs of two elements to exclude from the range, and we must make sure we don’t over count these. It’s PIE time! Example 1.6.8 How many functions f : {1, 2, 3, 4, 5} → {a, b, c} are surjective? Solution. Again start with the total number of functions: 35 (as each of the ﬁve elements of the domain can go to any of three elements of the codomain). Now we count the functions which are not surjective. Start by excluding a from the range. Then we have two choices (b or c) for where to send each of the ﬁve elements of the domain. Thus there are 25 functions which exclude a from the range. Similarly, there are 25 functions which exclude b, and another 25 which exclude c. Now have we counted all functions which are not surjective? Yes, but in fact, we have counted some multiple times. For example, the function which sends everything to c was one of the 25 functions we counted when we excluded a from the range, and also 120 1. Counting one of the 25 functions we counted when we excluded b from the range. We must subtract out all the functions which speciﬁcally exclude two elements from the range. There is 1 function when we exclude a and b (everything goes to c), one function when we exclude a and c, and one function when we exclude b and c. We are using PIE: to count the functions which are not surjective, we added up the functions which exclude a, b, and c separately, then subtracted the functions which exclude pairs of elements. We would then add back in the functions which exclude groups of three elements, except that there are no such functions. We ﬁnd that the number of functions which are not surjective is 25 + 25 + 25 − 1 − 1 − 1 + 0. Perhaps a more descriptive way to write this is 3 5 3 5 3 5 2 − 1 + 0 . 1 2 3 since each of the 25 ’s was the result of choosing 1 of the 3 elements of the codomain to exclude from the range, each of the three 15 ’s was the result of choosing 2 of the 3 elements of the codomain to exclude. Writing 15 instead of 1 makes sense too: we have 1 choice of where to send each of the 5 elements of the domain. Now we can ﬁnally count the number of surjective functions: 3 − 5 3 5 3 5 2 − 1 150. 1 2 You might worry that to count surjective functions when the codomain is larger than 3 elements would be too tedious. We need to use PIE but with more than 3 sets the formula for PIE is very long. However, we have lucked out. As we saw in the example above, the number of functions which exclude a single element from the range is the same no matter which single element is excluded. Similarly, the number of functions which exclude a pair of elements will be the same for every pair. With larger codomains, we will see the same behavior with groups of 3, 4, and more elements excluded. So instead of adding/subtracting each of these, we can simply add or subtract all of them at once, if you know how many there are. This works just like it did in for the other types of counting questions in this section, only now the size of the various combinations of sets is a number raised to a power, as opposed to a binomial coeﬃcient or factorial. Here’s what happens with 4 and 5 elements in the codomain. 1.6. Advanced Counting Using PIE 121 Example 1.6.9 1. How many functions f : {1, 2, 3, 4, 5} → {a, b, c, d} are surjective? 2. How many functions f : {1, 2, 3, 4, 5} → {a, b, c, d, e} are surjective? Solution. 1. There are 45 functions all together; we will subtract the functions which are not surjective. We could exclude any one of the four elements of the codomain, and doing so will leave us with 35 functions for each excluded element. This counts too many so we subtract the functions which exclude two of the four elements of the codomain, each pair giving 25 functions. But this excludes too many, so we add back in the functions which exclude three of the four elements of the codomain, each triple giving 15 function. There are 41 groups of functions excluding a single element, 42 groups of functions excluding a pair of elements, and 43 groups of functions excluding a triple of elements. This means that the number of functions which are not surjective is: 4 5 4 5 4 5 3 − 2 + 1 . 1 2 3 We can now say that the number of functions which are surjective is: 4 5 4 5 4 5 5 4 − 3 − 2 + 1 . 1 2 3 2. The number of surjective functions is: 5 − 5 5 5 5 5 5 5 5 5 4 − 3 + 2 − 1 . 1 2 3 4 We took the total number of functions 55 and subtracted all that were not surjective. There were 51 ways to select a single element from the codomain to exclude from the range, and for each there were 45 functions. But this double counts, so we use PIE and subtract functions excluding two elements 5 from the range: there are 2 choices for the two elements to exclude, and for each pair, 35 functions. This takes out too 122 1. Counting many functions, so we add back in functions which exclude 3 elements from the range: 53 choices for which three to exclude, and then 25 functions for each choice of elements. Finally we take back out the 1 function which excludes 4 elements for each of the 54 choices of 4 elements. If you happen to calculate this number precisely, you will get 120 surjections. That happens to also be the value of 5!. This might seem like an amazing coincidence until you realize that every surjective function f : X → Y with |X| |Y| ﬁnite must necessarily be a bĳection. The number of bĳections is always |X|! in this case. What we have here is a combinatorial proof of the following identity: n − n n n n (n − 1)n − (n − 2)n + · · · + 1n n!. 1 2 n−1 We have seen that counting surjective functions is another nice example of the advanced use of the Principle of Inclusion/Exclusion. Also, counting injective functions turns out to be equivalent to permutations, and counting all functions has a solution akin to those counting problems where order matters but repeats are allowed (like counting the number of words you can make from a given set of letters). These are not just a few more examples of the techniques we have developed in this chapter. Quite the opposite: everything we have learned in this chapter are examples of counting functions! Example 1.6.10 How many 5-letter words can you make using the eight letters a through h? How many contain no repeated letters? Solution. By now it should be no surprise that there are 85 words, and P(8, 5) words without repeated letters. The new piece here is that we are actually counting functions. For the ﬁrst problem, we are counting all functions from {1, 2, . . . , 5} to {a, b, . . . , h}. The numbers in the domain represent the position of the letter in the word, the codomain represents the letter that could be assigned to that position. If we ask for no repeated letters, we are asking for injective functions. If A and B are any sets with |A| 5 and |B| 8, then the number of functions f : A → B is 85 and the number of injections is P(8, 5). So if you can represent your counting problem as a function counting problem, most of the work is done. 123 1.6. Advanced Counting Using PIE Example 1.6.11 How many subsets are there of {1, 2, . . . , 9}? How many 9-bit strings are there (of any weight)? Solution. We saw in Section 1.2 that the answer to both these questions is 29 , as we can say yes or no (or 0 or 1) to each of the 9 elements in the set (positions in the bit-string). But 29 also looks like the answer you get from counting functions. In fact, if you count all functions f : A → B with |A| 9 and |B| 2, this is exactly what you get. This makes sense! Let A {1, 2, . . . , 9} and B { y, n}. We are assigning each element of the set either a yes or a no. Or in the language of bit-strings, we would take the 9 positions in the bit string as our domain and the set {0, 1} as the codomain. So far we have not used a function as a model for binomial coeﬃcients (combinations). Think for a moment about the relationship between 9 combinations and permutations, say speciﬁcally 3 and P(9, 3). We do have a function model for P(9, 3). This is the number of injective functions from a set of size 3 (say {1, 2, 3} to a set of size 9 (say {1, 2, . . . , 9}) since there are 9 choices for where to send the ﬁrst element of the domain, then only 8 choices for the second, and 7 choices for the third. For example, the function might look like this: f (1) 5 f (2) 8 f (3) 4. This is a diﬀerent function from: f (1) 4 f (2) 5 f (3) 8. Now P(9, 3) counts these as diﬀerent outcomes correctly, but 93 will count these (among others) as just one outcome. In fact, in terms of func9 tions 3 just counts the number of diﬀerent ranges possible of injective functions. This should not be a surprise since binomial coeﬃcients counts subsets, and the range is a possible subset of the codomain.3 While it is possible to interpret combinations as functions, perhaps the better advice is to instead use combinations (or stars and bars) when functions are not quite the right way to interpret the counting question. 3A more mathematically sophisticated interpretation of combinations is that we are deﬁning two injective functions to be equivalent if they have the same range, and then counting the number of equivalence classes under this notion of equivalence. 124 1. Counting Exercises 1. The dollar menu at your favorite tax-free fast food restaurant has 7 items. You have $10 to spend. How many diﬀerent meals can you buy if you spend all your money and: (a) Purchase at least one of each item. (b) Possibly skip some items. (c) Don’t get more than 2 of any particular item. 2. After a late night of math studying, you and your friends decide to go to your favorite tax-free fast food Mexican restaurant, Burrito Chime. You decide to order oﬀ of the dollar menu, which has 7 items. Your group has $16 to spend (and will spend all of it). (a) How many diﬀerent orders are possible? Explain. (The order in which the order is placed does not matter - just which and how many of each item that is ordered.) (b) How many diﬀerent orders are possible if you want to get at least one of each item? Explain. (c) How many diﬀerent orders are possible if you don’t get more than 4 of any one item? Explain. 3. After another gym class you are tasked with putting the 14 identical dodgeballs away into 5 bins. This time, no bin can hold more than 6 balls. How many ways can you clean up? 4. Consider the equation x 1 + x 2 + x 3 + x 4 15. How many solutions are there with 2 ≤ x i ≤ 5 for all i ∈ {1, 2, 3, 4}? 5. Suppose you planned on giving 7 gold stars to some of the 13 star students in your class. Each student can receive at most one star. How many ways can you do this? Use PIE. Then, ﬁnd the numeric answer in Pascal’s triangle and explain why that makes sense. 6. Based on the previous question, give a combinatorial proof for the identity: Õ n n+k−1 n − (−1) j+1 k k j n j1 7. n + k − (2 j + 1) . k − 2j Illustrate how the counting of derangements works by writing all permutations of {1, 2, 3, 4} and the crossing out those which are not derangements. Keep track of the permutations you cross out more than once, using PIE. 1.6. Advanced Counting Using PIE 125 8. How many permutations of {1, 2, 3, 4, 5} leave exactly 1 element ﬁxed? 9. Ten ladies of a certain age drop oﬀ their red hats at the hat check of a museum. As they are leaving, the hat check attendant gives the hats back randomly. In how many ways can exactly six of the ladies receive their own hat (and the other four not)? Explain. 10. The Grinch sneaks into a room with 6 Christmas presents to 6 diﬀerent people. He proceeds to switch the name-labels on the presents. How many ways could he do this if: (a) No present is allowed to end up with its original label? Explain what each term in your answer represents. (b) Exactly 2 presents keep their original labels? Explain. (c) Exactly 5 presents keep their original labels? Explain. 11. Consider functions f : {1, 2, 3, 4} → {a, b, c, d, e , f }. How many functions have the property that f (1) , a or f (2) , b, or both? 12. Consider sets A and B with |A| 10 and |B| 5. How many functions f : A → B are surjective? 13. Let A {1, 2, 3, 4, 5}. How many injective functions f : A → A have the property that for each x ∈ A, f (x) , x? 14. Let d n be the number of derangements of n objects. For example, using the techniques of this section, we ﬁnd d3 3! − 3 3 3 2! − 1! + 0! . 1 2 3 We can use the formula for nk to write this all in terms of factorials. After simplifying, for d3 we would get 1 1 1 d3 3! 1 − + − . 1 2 6 Generalize this to ﬁnd a nicer formula for d n . Bonus: For large n, approximately what fraction of all permutations are derangements? Use your knowledge of Taylor series from calculus. 126 1. Counting 1.7. Chapter Summary 1.7 127 Chapter Summary Investigate! Suppose you have a huge box of animal crackers containing plenty of each of 10 diﬀerent animals. For the counting questions below, carefully examine their similarities and diﬀerences, and then give an answer. The answers are all one of the following: 10 15 6 P(10, 6) 10 6 9 . 1. How many animal parades containing 6 crackers can you line up? 2. How many animal parades of 6 crackers can you line up so that the animals appear in alphabetical order? 3. How many ways could you line up 6 diﬀerent animals in alphabetical order? 4. How many ways could you line up 6 diﬀerent animals if they can come in any order? 5. How many ways could you give 6 children one animal cracker each? 6. How many ways could you give 6 children one animal cracker each so that no two kids get the same animal? 7. How many ways could you give out 6 giraﬀes to 10 kids? 8. Write a question about giving animal crackers to kids that has the answer 10 . 6 ! Attempt the above activity before proceeding ! With all the diﬀerent counting techniques we have mastered in this last chapter, it might be diﬃcult to know when to apply which technique. Indeed, it is very easy to get mixed up and use the wrong counting method for a given problem. You get better with practice. As you practice you start to notice some trends that can help you distinguish between types of counting problems. Here are some suggestions that you might ﬁnd helpful when deciding how to tackle a counting problem and checking whether your solution is correct. • Remember that you are counting the number of items in some list of outcomes. Write down part of this list. Write down an element in the middle of the list – how are you deciding whether your element 128 1. Counting really is in the list. Could you get this element more than once using your proposed answer? • If generating an element on the list involves selecting something (for example, picking a letter or picking a position to put a letter, etc), can the things you select be repeated? Remember, permutations and combinations select objects from a set without repeats. • Does order matter? Be careful here and be sure you know what your answer really means. We usually say that order matters when you get diﬀerent outcomes when the same objects are selected in diﬀerent orders. Combinations and “Stars & Bars” are used when order does not matter. • There are four possibilities when it comes to order and repeats. If order matters and repeats are allowed, the answer will look like n k . If order matters and repeats are not allowed, we have P(n, k). If order doesn’t matter and repeats are allowed, use stars and bars. If order doesn’t matter and repeats are not allowed, use nk . But be careful: this only applies when you are selecting things, and you should make sure you know exactly what you are selecting before determining which case you are in. • Think about how you would represent your counting problem in terms of sets or functions. We know how to count diﬀerent sorts of sets and diﬀerent types of functions. • As we saw with combinatorial proofs, you can often solve a counting problem in more than one way. Do that, and compare your numerical answers. If they don’t match, something is amiss. While we have covered many counting techniques, we have really only scratched the surface of the large subject of enumerative combinatorics. There are mathematicians doing original research in this area even as you read this. Counting can be really hard. In the next chapter, we will approach counting questions from a very diﬀerent direction, and in doing so, answer inﬁnitely many counting questions at the same time. We will create sequences of answers to related questions. Chapter Review 1. You have 9 presents to give to your 4 kids. How many ways can this be done if: (a) The presents are identical, and each kid gets at least one present? (b) The presents are identical, and some kids might get no presents? 1.7. Chapter Summary 129 (c) The presents are unique, and some kids might get no presents? (d) The presents are unique and each kid gets at least one present? 2. For each of the following counting problems, say whether the answer 10 is 4 , P(10, 4), or neither. If you answer is “neither,” say what the answer should be instead. (a) How many shortest lattice paths are there from (0, 0) to (10, 4)? (b) If you have 10 bow ties, and you want to select 4 of them for next week, how many choices do you have? (c) Suppose you have 10 bow ties and you will wear a diﬀerent one on each of the next 4 days. How many choices do you have? (d) If you want to wear 4 of your 10 bow ties next week (Monday through Sunday), how many ways can this be accomplished? (e) Out of a group of 10 classmates, how many ways can you rank your top 4 friends? (f) If 10 students come to their professor’s oﬃce but only 4 can ﬁt at a time, how diﬀerent combinations of 4 students can see the prof ﬁrst? (g) How many 4 letter words can be made from the ﬁrst 10 letters of the alphabet? (h) How many ways can you make the word “cake” from the ﬁrst 10 letters of the alphabet? (i) How many ways are there to distribute 10 identical apples among 4 children? (j) If you have 10 kids (and live in a shoe) and 4 types of cereal, how many ways can your kids eat breakfast? (k) How many ways can you arrange exactly 4 ones in a string of 10 binary digits? (l) You want to select 4 distinct, single-digit numbers as your lotto picks. How many choices do you have? (m) 10 kids want ice-cream. You have 4 varieties. How many ways are there to give the kids as much ice-cream as they want? (n) How many 1-1 functions are there from {1, 2, . . . , 10} to {a, b, c, d}? (o) How many surjective functions are there from {1, 2, . . . , 10} to {a, b, c, d}? (p) Each of your 10 bow ties match 4 pairs of suspenders. How many outﬁts can you make? 130 1. Counting (q) After the party, the 10 kids each choose one of 4 party-favors. How many outcomes? (r) How many 6-elements subsets are there of the set {1, 2, . . . , 10} (s) How many ways can you split up 11 kids into 5 named teams? (t) How many solutions are there to x1 + x2 + · · · + x 5 6 where each x i is a non-negative integer? (u) Your band goes on tour. There are 10 cities within driving distance, but only enough time to play 4 of them. How many choices do you have for the cities on your tour? (v) In how many diﬀerent ways can you play the 4 cities you choose? (w) Out of the 10 breakfast cereals available, you want to have 4 bowls. How many ways can you do this? (x) There are 10 types of cookies available. You want to make a 4 cookie stack. How many diﬀerent stacks can you make? (y) From your home at (0,0) you want to go to either the donut shop at (5,4) or the one at (3,6). How many paths could you take? (z) How many 10-digit numbers do not contain a sub-string of 4 repeated digits? 3. bow tiesRecall, you own 3 regular ties and 5 bow ties. You realize that it would be okay to wear more than two ties to your clown college interview. (a) You must select some of your ties to wear. Everything is okay, from no ties up to all ties. How many choices do you have? (b) If you want to wear at least one regular tie and one bow tie, but are willing to wear up to all your ties, how many choices do you have for which ties to wear? (c) How many choices of which ties to wear do you have if you wear exactly 2 of the 3 regular ties and 3 of the 5 bow ties? (d) Once you have selected 2 regular and 3 bow ties, in how many orders could you put the ties on, assuming you must have one of the three bow ties on top? 4. Give a counting question where the answer is 8 · 3 · 3 · 5. Give another question where the answer is 8 + 3 + 3 + 5. 5. Consider ﬁve digit numbers α a 1 a 2 a 3 a4 a 5 , with each digit from the set {1, 2, 3, 4}. (a) How many such numbers are there? 1.7. Chapter Summary 131 (b) How many such numbers are there for which the sum of the digits is even? (c) How many such numbers contain more even digits than odd digits? 6. In a recent small survey of airline passengers, 25 said they had ﬂown American in the last year, 30 had ﬂown Jet Blue, and 20 had ﬂown Continental. Of those, 10 reported they had ﬂown on American and Jet Blue, 12 had ﬂown on Jet Blue and Continental, and 7 had ﬂown on American and Continental. 5 passengers had ﬂown on all three airlines. How many passengers were surveyed? (Assume the results above make up the entire survey.) 7. Recall, by 8-bit strings, we mean strings of binary digits, of length 8. (a) How many 8-bit strings are there total? (b) How many 8-bit strings have weight 5? (c) How many subsets of the set {a, b, c, d, e , f , g, h} contain exactly 5 elements? (d) Explain why your answers to parts (b) and (c) are the same. Why are these questions equivalent? 8. What is the coeﬃcient of x 10 in the expansion of (x + 1)13 + x 2 (x + 1)17 ? 9. How many 8-letter words contain exactly 5 vowels? (One such word is “aaioobtt”; don’t consider “y” a vowel for this exercise.) What if repeated letters were not allowed? 10. For each of the following, ﬁnd the number of shortest lattice paths from (0, 0) to (8, 8) which: (a) pass through the point (2, 3). (b) avoid (do not pass through) the point (7, 5). (c) either pass through (2, 3) or (5, 7) (or both). 11. You live in Grid-Town on the corner of 2nd and 3rd, and work in a building on the corner of 10th and 13th. How many routes are there which take you from home to work and then back home, but by a diﬀerent route? 12. How many 10-bit strings start with 111 or end with 101 or both? 13. How many 10-bit strings of weight 6 start with 111 or end with 101 or both? 132 1. Counting 14. How many 6 letter words made from the letters a, b, c, d, e , f without repeats do not contain the sub-word “bad” in consecutive letters? How many don’t to contain the subword “bad” in not-necessarily consecutive letters (but in order)? 15. Explain using lattice paths why n k0 k Ín 2n . 16. Suppose you have 20 one-dollar bills to give out as prizes to your top 5 discrete math students. How many ways can you do this if: (a) Each of the 5 students gets at least 1 dollar? (b) Some students might get nothing? (c) Each student gets at least 1 dollar but no more than 7 dollars? 17. How many functions f : {1, 2, 3, 4, 5} → {a, b, c, d, e} are there satisfying: (a) f (1) a or f (2) b (or both)? (b) f (1) , a or f (2) , b (or both)? (c) f (1) , a and f (2) , b, and f is injective? (d) f is surjective, but f (1) , a, f (2) , b, f (3) , c, f (4) , d and f (5) , e? 18. How many functions map {1, 2, 3, 4, 5, 6} onto {a, b, c, d} (i.e., how many surjections are there)? 19. To thank your math professor for doing such an amazing job all semester, you decide to bake Oscar cookies. You know how to make 10 diﬀerent types of cookies. (a) If you want to give your professor 4 diﬀerent types of cookies, how many diﬀerent combinations of cookie type can you select? Explain your answer. (b) To keep things interesting, you decide to make a diﬀerent number of each type of cookie. If again you want to select 4 cookie types, how many ways can you select the cookie types and decide for which there will be the most, second most, etc. Explain your answer. (c) You change your mind again. This time you decide you will make a total of 12 cookies. Each cookie could be any one of the 10 types of cookies you know how to bake (and it’s okay if you leave some types out). How many choices do you have? Explain. 1.7. Chapter Summary 133 (d) You realize that the previous plan did not account for presentation. This time, you once again want to make 12 cookies, each one could be any one of the 10 types of cookies. However, now you plan to shape the cookies into the numerals 1, 2, ..., 12 (and probably arrange them to make a giant clock, but you haven’t decided on that yet). How many choices do you have for which types of cookies to bake into which numerals? Explain. (e) The only ﬂaw with the last plan is that your professor might not get to sample all 10 diﬀerent varieties of cookies. How many choices do you have for which types of cookies to make into which numerals, given that each type of cookie should be present at least once? Explain. 20. For which of the parts of the previous problem (Exercise 1.7.19) does it make sense to interpret the counting question as counting some number of functions? Say what the domain and codomain should be, and whether you are counting all functions, injections, surjections, or something else. 134 1. Counting Chapter 2 Sequences Investigate! There is a monastery in Hanoi, as the legend goes, with a great hall containing three tall pillars. Resting on the ﬁrst pillar are 64 giant disks (or washers), all diﬀerent sizes, stacked from largest to smallest. The monks are charged with the following task: they must move the entire stack of disks to the third pillar. However, due to the size of the disks, the monks cannot move more than one at a time. Each disk must be placed on one of the pillars before the next disk is moved. And because the disks are so heavy and fragile, the monks may never place a larger disk on top of a smaller disk. When the monks ﬁnally complete their task, the world shall come to an end. Your task: ﬁgure out how long before we need to start worrying about the end of the world. 1. First, let’s ﬁnd the minimum number of moves required for a smaller number of disks. Collect some data. Make a table. 2. Conjecture a formula for the minimum number of moves required to move n disks. Test your conjecture. How do you know your formula is correct? 3. If the monks were able to move one disk every second without ever stopping, how long before the world ends? ! Attempt the above activity before proceeding ! This puzzle is called the Tower of Hanoi. You are tasked with ﬁnding the minimum number of moves to complete the puzzle. This certainly sounds like a counting problem. Perhaps you have an answer? If not, what else could we try? The answer depends on the number of disks you need to move. In fact, we could answer the puzzle ﬁrst for 1 disk, then 2, then 3 and so on. If we list out all of the answers for each number of disks, we will get a sequence of numbers. The nth term in the sequence is the answer to the question, “what is the smallest number of moves required to complete the Tower of Hanoi puzzle with n disks?” You might wonder why we would create such a sequence instead of just answering the question. By looking at how the sequence of numbers grows, we gain insight into the problem. It is easy to count the number of moves required for a small number of disks. We can then look for a pattern among the ﬁrst few terms 135 136 2. Sequences of the sequence. Hopefully this will suggest a method for ﬁnding the nth term of the sequence, which is the answer to our question. Of course we will also need to verify that our suspected pattern is correct, and that this correct pattern really does give us the nth term we think it does, but it is impossible to prove that your formula is correct without having a formula to start with. Sequences are also interesting mathematical objects to study in their own right. Let’s see why. 2.1 Describing Sequences Investigate! You have a large collection of 1 × 1 squares and 1 × 2 dominoes. You want to arrange these to make a 1 × 15 strip. How many ways can you do this? 1. Start by collecting data. How many length 1 × 1 strips can you make? How many 1 × 2 strips? How many 1 × 3 strips? And so on. 2. How are the 1 × 3 and 1 × 4 strips related to the 1 × 5 strips? 3. How many 1 × 15 strips can you make? 4. What if I asked you to ﬁnd the number of 1 × 1000 strips? Would the method you used to calculate the number fo 1×15 strips be helpful? ! Attempt the above activity before proceeding ! A sequence is simply an ordered list of numbers. For example, here is a sequence: 0, 1, 2, 3, 4, 5, . . . . This is diﬀerent from the set N because, while the sequence is a complete list of every element in the set of natural numbers, in the sequence we very much care what order the numbers come in. For this reason, when we use variables to represent terms in a sequence they will look like this: a0 , a1 , a2 , a3 , . . . . To refer to the entire sequence at once, we will write (a n )n∈N or (a n )n≥0 , or sometimes if we are being sloppy, just (a n ) (in which case we assume we start the sequence with a 0 ). We might replace the a with another letter, and sometimes we omit a 0 , starting with a 1 , in which case we would use (a n )n≥1 to refer to the 2.1. Describing Sequences 137 sequence as a whole. The numbers in the subscripts are called indices (the plural of index). While we often just think of a sequence as an ordered list of numbers, it is really a type of function. Speciﬁcally, the sequence (a n )n≥0 is a function with domain N where a n is the image of the natural number n. Later we will manipulate sequences in much the same way you have manipulated functions in algebra or calculus. We can shift a sequence up or down, add two sequences, or ask for the rate of change of a sequence. These are done exactly as you would for functions. That said, while keeping the rigorous mathematical deﬁnition in mind is helpful, we often describe sequences by writing out the ﬁrst few terms. Example 2.1.1 Can you ﬁnd the next term in the following sequences? 1. 7, 7, 7, 7, 7, . . . 2. 3, −3, 3, −3, 3, . . . 3. 1, 5, 2, 10, 3, 15, . . . 4. 1, 2, 4, 8, 16, 32, . . . 5. 1, 4, 9, 16, 25, 36, . . . 6. 1, 2, 3, 5, 8, 13, 21, . . . 7. 1, 3, 6, 10, 15, 21, . . . 8. 2, 3, 5, 7, 11, 13, . . . 9. 3, 2, 1, 0, −1, . . . 10. 1, 1, 2, 6, . . . Solution. No you cannot. You might guess that the next terms are: 1. 7 4. 64 7. 28 2. −3 5. 49 8. 17 3. 4 6. 34 9. −2 10. 24 In fact, those are the next terms of the sequences I had in mind when I made up the example, but there is no way to be sure they are correct. Still, we will often do this. Given the ﬁrst few terms of a sequence, we can ask what the pattern in the sequence suggests the next terms are. 138 2. Sequences Given that no number of initial terms in a sequence is enough to say for certain which sequence we are dealing with, we need to ﬁnd another way to specify a sequence. We consider two ways to do this: Closed formula. A closed formula for a sequence (a n )n∈N is a formula for a n using a ﬁxed ﬁnite number of operations on n. This is what you normally think of as a formula in n, just as if you were deﬁning a function in terms of n (because that is exactly what you are doing). Recursive deﬁnition. A recursive deﬁnition (sometimes called an inductive deﬁnition) for a sequence (a n )n∈N consists of a recurrence relation : an equation relating a term of the sequence to previous terms (terms with smaller index) and an initial condition: a list of a few terms of the sequence (one less than the number of terms in the recurrence relation). It is easier to understand what is going on here with an example: Example 2.1.2 Here are a few closed formulas for sequences: • an n2. • an n(n + 1) . 2 • an √ n 1+ 5 2 − √ √ −n 1− 5 2 5 . Note in each formula, if you are given n, you can calculate a n directly: just plug in n. For example, to ﬁnd a3 in the second 3(3+1) sequence, just compute a 3 2 6. Here are a few recursive deﬁnitions for sequences: • a n 2a n−1 with a 0 1. • a n 2a n−1 with a 0 27. • a n a n−1 + a n−2 with a 0 0 and a 1 1. In these formulas, if you are given n, you cannot calculate a n directly, you ﬁrst need to ﬁnd a n−1 (or a n−1 and a n−2 ). In the second sequence, to ﬁnd a 3 you would take 2a 2 , but to ﬁnd a 2 2a 1 we would need to know a 1 2a0 . We do know this, so we could trace 2.1. Describing Sequences 139 back through these equations to ﬁnd a 1 54, a 2 108 and ﬁnally a 3 216. You might wonder why we would bother with recursive deﬁnitions for sequences. After all, it is harder to ﬁnd a n with a recursive deﬁnition than with a closed formula. This is true, but it is also harder to ﬁnd a closed formula for a sequence than it is to ﬁnd a recursive deﬁnition. So to ﬁnd a useful closed formula, we might ﬁrst ﬁnd the recursive deﬁnition, then use that to ﬁnd the closed formula. This is not to say that recursive deﬁnitions aren’t useful in ﬁnding a n . You can always calculate a n given a recursive deﬁnition, it might just take a while. Example 2.1.3 Find a 6 in the sequence deﬁned by a n 2a n−1 − a n−2 with a 0 3 and a 1 4. Solution. We know that a 6 2a5 − a 4 . So to ﬁnd a6 we need to ﬁnd a 5 and a 4 . Well a 5 2a 4 − a 3 and a 4 2a 3 − a 2 , so if we can only ﬁnd a 3 and a 2 we would be set. Of course a 3 2a 2 − a 1 and a 2 2a 1 − a 0 , so we only need to ﬁnd a 1 and a 0 . But we are given these. Thus a0 3 a1 4 a2 2 · 4 − 3 5 a3 2 · 5 − 4 6 a4 2 · 6 − 5 7 a5 2 · 7 − 6 8 a 6 2 · 8 − 7 9. Note that now we can guess a closed formula for the nth term of the sequence: a n n + 3. To be sure this will always work, we could plug in this formula into the recurrence relation: 2a n−1 − a n−2 2((n − 1) + 3) − ((n − 2) + 3) 2n + 4 − n − 1 n + 3 an . 140 2. Sequences That is not quite enough though, since there can be multiple closed formulas that satisfy the same recurrence relation; we must also check that our closed formula agrees on the initial terms of the sequence. Since a 0 0 + 3 3 and a 1 1 + 3 4 are the correct initial conditions, we can now conclude we have the correct closed formula. Finding closed formulas, or even recursive deﬁnitions, for sequences is not trivial. There is no one method for doing this. Just as in evaluating integrals or solving diﬀerential equations, it is useful to have a bag of tricks you can apply, but sometimes there is no easy answer. One useful method is to relate a given sequence to another sequence for which we already know the closed formula. To do this, we need a few “known sequences” to compare mystery sequences to. Here are a few that are good to know. We will verify the formulas for these in the coming sections. Common Sequences. 1, 4, 9, 16, 25, . . . The square numbers. The sequence (s n )n≥1 has closed formula s n n 2 1, 3, 6, 10, 15, 21, . . . The triangular numbers. The sequence (Tn )n≥1 has closed n(n+1) formula Tn 2 . 1, 2, 4, 8, 16, 32, . . . The powers of 2. The sequence (a n )n≥0 with closed formula a n 2n . 1, 1, 2, 3, 5, 8, 13, . . . The Fibonacci numbers (or Fibonacci sequence), deﬁned recursively by Fn Fn−1 + Fn−2 with F1 F2 1 Example 2.1.4 n(n+1) Use the formulas Tn 2 and a n 2n to ﬁnd closed formulas that agree with the following sequences. Assume each ﬁrst term corresponds to n 0. 1. (b n ): 1, 2, 4, 7, 11, 16, 22, . . .. 2. (c n ): 3, 5, 9, 17, 33, . . .. 2.1. Describing Sequences 141 3. (d n ): 0, 2, 6, 12, 20, 30, 42, . . .. 4. (e n ): 3, 6, 10, 15, 21, 28, . . .. 5. ( f n ): 0, 1, 3, 7, 15, 31, . . .. 6. (g n ) 3, 6, 12, 24, 48, . . .. 7. (h n ): 6, 10, 18, 34, 66, . . .. 8. (jn ): 15, 33, 57, 87, 123, . . .. Solution. We wish to compare these sequences to the triangular numbers (0, 1, 3, 6, 10, 15, 21, . . .), when we start with n 0, and the powers of 2: (1, 2, 4, 8, 16, . . .). 1. (1, 2, 4, 7, 11, 16, 22, . . .). Note that if subtract 1 from each term, we get the sequence (Tn ). So we have b n Tn + 1. n(n+1) Therefore a closed formula is b n 2 + 1. A quick check of the ﬁrst few n conﬁrms we have it right. 2. (3, 5, 9, 17, 33, . . .). Each term in this sequence is one more than a power of 2, so we might guess the closed formula is c n a n +1 2n +1. If we try this though, we get c0 20 +1 2 and c 1 21 + 1 3. We are oﬀ because the indices are shifted. What we really want is c n a n+1 + 1 giving c n 2n+1 + 1. 3. (0, 2, 6, 12, 20, 30, 42, . . .). Notice that all these terms are even. What happens if we factor out a 2? We get (Tn )! More precisely, we ﬁnd that d n /2 Tn , so this sequence has closed formula d n n(n + 1). 4. (3, 6, 10, 15, 21, 28, . . .). These are all triangular numbers. However, we are starting with 3 as our initial term instead of as our third term. So if we could plug in 2 instead of 0 into the formula for Tn , we would be set. Therefore the closed (n+2)(n+3) (where n + 3 came from (n + 2) + 1). formula is e n 2 Thinking about sequences as functions, we are doing a horizontal shift by 2: e n Tn+2 which would cause the graph to shift 2 units to the left. 5. (0, 1, 3, 7, 15, 31, . . .). Try adding 1 to each term and we get powers of 2. You might guess this because each term is a little more than twice the previous term (the powers of 2 are exactly twice the previous term). Closed formula: f n 2n − 1. 6. (3, 6, 12, 24, 48, . . .). These numbers are also doubling each time, but are also all multiples of 3. Dividing each by 3 gives 1, 2, 4, 8, . . . . Aha. We get the closed formula g n 3 · 2n . 142 2. Sequences 7. (6, 10, 18, 34, 66, . . .). To get from one term to the next, we almost double each term. So maybe we can relate this back to 2n . Yes, each term is 2 more than a power of 2. So we get h n 2n+2 + 2 (the n + 2 is because the ﬁrst term is 2 more than 22 , not 20 ). Alternatively, we could have related this sequence to the second sequence in this example: starting with 3, 5, 9, 17, . . . we see that this sequence is twice the terms from that sequence. That sequence had closed formula c n 2n+1 + 1. Our sequence here would be twice this, so h n 2(2n + 1), which is the same as we got before. 8. (15, 33, 57, 87, 123, . . .). Try dividing each term by 3. That gives the sequence 5, 11, 19, 29, 41, . . .. Now add 1 to each term: 6, 12, 20, 30, 42, . . ., which is (d n ) in this example, except starting with 6 instead of 0. So let’s start with the formula d n n(n + 1). To start with the 6, we shift: (n + 2)(n + 3). But this is one too many, so subtract 1: (n + 2)(n + 3) − 1. That gives us our sequence, but divided by 3. So we want jn 3((n + 2)(n + 3) − 1). Partial sums. Some sequences naturally arise as the sum of terms of another sequence. Example 2.1.5 Sam keeps track of how many push-ups she does each day of her “do lots of push-ups challenge.” Let (a n )n≥1 be the sequence that describes the number of push-ups done on the nth day of the challenge. The sequence starts 3, 5, 6, 10, 9, 0, 12, . . . . Describe a sequence (b n )n≥1 that describes the total number of push-ups done by Sam after the nth day. Solution. We can ﬁnd the terms of this sequence easily enough. 3, 8, 14, 24, 33, 33, 45, . . . . Here b 1 is just a 1 , but then b2 3 + 5 a1 + a2 , b3 3 + 5 + 6 a1 + a2 + a3 , 2.1. Describing Sequences 143 and so on. There are a few ways we might describe b n in general. We could do so recursively as, b n b n−1 + a n , since the total number of push-ups done after n days will be the number done after n − 1 days, plus the number done on day n. For something closer to a closed formula, we could write b n a1 + a2 + a3 + · · · + a n , or the same thing using summation notation: bn n Õ ai . i1 However, note that these are not really closed formulas since even if we had a formula for a n , we would still have an increasing number of computations to do as n increases. Given any sequence (a n )n∈N , we can always form a new sequence (b n )n∈N by b n a0 + a1 + a2 + · · · + a n . Since the terms of (b n ) are the sums of the initial part of the sequence (a n ) ways call (b n ) the sequence of partial sums of (a n ). Soon we will see that it is sometimes possible to ﬁnd a closed formula for (b n ) from the closed formula for (a n ). To simplify writing out these sums, we will often use notation like n Õ a k . This means add up the a k ’s where k changes from 1 to n. k1 Example 2.1.6 Use Í notation to rewrite the sums: 1. 1 + 2 + 3 + 4 + · · · + 100 2. 1 + 2 + 4 + 8 + · · · + 250 3. 6 + 10 + 14 + · · · + (4n − 2). 144 2. Sequences Solution. 1. 100 Õ k 2. k1 50 Õ 2 k k0 3. n Õ (4k − 2) k2 If we want to multiply the a k instead, we could write example, n Ö n Ö a k . For k1 k n!. k1 Exercises 1. Find the closed formula for each of the following sequences by relating them to a well known sequence. Assume the ﬁrst term given is a 1 . (a) 2, 5, 10, 17, 26, . . . (b) 0, 2, 5, 9, 14, 20, . . . (c) 8, 12, 17, 23, 30, . . . (d) 1, 5, 23, 119, 719, . . . 2. For each sequence given below, ﬁnd a closed formula for a n , the nth term of the sequence (assume the ﬁrst terms are a0 ) by relating it to another sequence for which you already know the formula. In each case, brieﬂy say how you got your answers. (a) 4, 5, 7, 11, 19, 35, . . . (b) 0, 3, 8, 15, 24, 35, . . . (c) 6, 12, 20, 30, 42, . . . (d) 0, 2, 7, 15, 26, 40, 57, . . . (Cryptic Hint: these might be called “house numbers”) 3. Write out the ﬁrst 5 terms (starting with a 0 ) of each of the sequences described below. Then give either a closed formula or a recursive deﬁnition for the sequence (whichever is NOT given in the problem). (a) a n 12 (n 2 + n). (b) a n 2a n−1 − a n−2 with a 0 0 and a 1 1. (c) a n na n−1 with a 0 1. 2.1. Describing Sequences 4. 145 Consider the sequence (a n )n≥1 that starts 1, 3, 5, 7, 9, . . . (i.e, the odd numbers in order). (a) Give a recursive deﬁnition and closed formula for the sequence. (b) Write out the sequence (b n )n≥2 of partial sums of (a n ). Write down the recursive deﬁnition for (b n ) and guess at the closed formula. 5. The Fibonacci sequence is 0, 1, 1, 2, 3, 5, 8, 13, . . . (where F0 0). (a) Write out the ﬁrst few terms of the sequence of partial sums: 0, 0 + 1, 0 + 1 + 1,. . . (b) Guess a formula for the sequence of partial sums expressed in terms of a single Fibonacci number. For example, you might say 2 F0 + F1 + · · · + Fn 3Fn−1 + n, although that is deﬁnitely not correct. 6. Consider the three sequences below. For each, ﬁnd a recursive deﬁnition. How are these sequences related? (a) 2, 4, 6, 10, 16, 26, 42, . . .. (b) 5, 6, 11, 17, 28, 45, 73, . . .. (c) 0, 0, 0, 0, 0, 0, 0, . . .. 7. Write out the ﬁrst few terms of the sequence given by a 1 3; a n 2a n−1 +4. Then ﬁnd a recursive deﬁnition for the sequence 10, 24, 52, 108, . . .. 8. Write out the ﬁrst few terms of the sequence given by a n n 2 − 3n + 1. Then ﬁnd a closed formula for the sequence (starting with a 1 ) 0, 2, 6, 12, 20, . . .. 9. Show that a n 3 · 2n + 7 · 5n is a solution to the recurrence relation a n 7a n−1 − 10a n−2 . What would the initial conditions need to be for this to be the closed formula for the sequence? 10. Show that a n 2n − 5n is also a solution to the recurrence relation a n 7a n−1 − 10a n−2 . What would the initial conditions need to be for this to be the closed formula for the sequence? 11. Find a closed formula for the sequence with recursive deﬁnition a n 2a n−1 − a n−2 with a 1 1 and a 2 2. 12. Give two diﬀerent recursive deﬁnitions for the sequence with closed formula a n 3 + 2n. Prove you are correct. At least one of the recursive deﬁnitions should makes use of two previous terms and no constants. 146 2. Sequences Í Î 13. Use summation ( ) or product ( ) notation to rewrite the following. (a) 2 + 4 + 6 + 8 + · · · + 2n. (d) 2 · 4 · 6 · · · · · 2n. (b) 1 + 5 + 9 + 13 + · · · + 425. (c) 1 + 1 2 + 1 3 + 1 4 +···+ 1 50 . 100 (e) ( 12 )( 23 )( 34 ) · · · ( 101 ). 14. Expand the following sums and products. That is, write them out the long way. (a) 100 Õ (3 + 4k). (d) k1 (b) n Õ k2 2k . (e) k0 (c) n Ö k2 . (k 2 − 1) (2 + 3k). k0 50 Õ k2 100 Ö (k 2 1 . − 1) 15. Suppose you draw n lines in the plane so that every pair of lines cross (no lines are parallel) and no three lines cross at the same point. This will create some number of regions in the plane, including some unbounded regions. Call the number of regions R n . Find a recursive formula for the number of regions created by n lines, and justify why your recursion is correct. 16. A ternary string is a sequence of 0’s, 1’s and 2’s. Just like a bit string, but with three symbols. Let’s call a ternary string good provided it never contains a 2 followed immediately by a 0. Let G n be the number of good strings of length n. For example, G1 3, and G2 8 (since of the 9 ternary strings of length 2, only one is not good). Find, with justiﬁcation, a recursive formula for G n , and use it to compute G5 . 17. Consider bit strings with length l and weight k (so strings of l 0’s and 1’s, including k 1’s). We know how to count the number of these for a ﬁxed l and k. Now, we will count the number of strings for which the sum of the length and the weight is ﬁxed. For example, let’s count all the bit strings for which l + k 11. (a) Find examples of these strings of diﬀerent lengths. What is the longest string possible? What is the shortest? (b) How many strings are there of each of these lengths. Use this to count the total number of strings (with sum 11). (c) The other approach: Let n l + k vary. How many strings have sum n 1? How many have sum n 2? And so on. Find and 2.1. Describing Sequences 147 explain a recurrence relation for the sequence (a n ) which gives the number of strings with sum n. (d) Describe what you have found above in terms of Pascal’s Triangle. What pattern have you discovered? 18. When bees play chess, they use a hexagonal board like the one shown below. The queen bee can move one space at a time either directly to the right or angled up-right or down-right (but can never move leftwards). How many diﬀerent paths can the queen take from the top left hexagon to the bottom right hexagon? Explain your answer, and this relates to the previous question. (As an example, there are three paths to get to the second hexagon on the bottom row.) start 3 stop 19. Let t n denote the number of ways to tile a 2×n checkerboard using 1×2 dominoes. Write out the ﬁrst few terms of the sequence (t n )n≥1 and then give a recursive deﬁnition. Explain why your recursive formula is correct. 148 2. Sequences 2.2 Arithmetic and Geometric Sequences Investigate! For the patterns of dots below, draw the next pattern in the sequence. Then give a recursive deﬁnition and a closed formula for the number of dots in the nth pattern. n0 n1 n2 n0 n1 n2 n1 ! n2 n3 n4 Attempt the above activity before proceeding ! We now turn to the question of ﬁnding closed formulas for particular types of sequences. Arithmetic Sequences. If the terms of a sequence diﬀer by a constant, we say the sequence is arithmetic. If the initial term (a 0 ) of the sequence is a and the common diﬀerence is d, then we have, Recursive deﬁnition: a n a n−1 + d with a 0 a. Closed formula: a n a + dn. How do we know this? For the recursive deﬁnition, we need to specify a 0 . Then we need to express a n in terms of a n−1 . If we call the ﬁrst term a, then a 0 a. For the recurrence relation, by the deﬁnition of an arithmetic sequence, the diﬀerence between successive terms is some constant, say d. So a n − a n−1 d, or in other words, a0 a a n a n−1 + d. 2.2. Arithmetic and Geometric Sequences 149 To ﬁnd a closed formula, ﬁrst write out the sequence in general: a0 a a1 a0 + d a + d a 2 a 1 + d a + d + d a + 2d a 3 a 2 + d a + 2d + d a + 3d .. . We see that to ﬁnd the nth term, we need to start with a and then add d a bunch of times. In fact, add it n times. Thus a n a + dn. Example 2.2.1 Find recursive deﬁnitions and closed formulas for the arithmetic sequences below. Assume the ﬁrst term listed is a 0 . 1. 2, 5, 8, 11, 14, . . .. 2. 50, 43, 36, 29, . . .. Solution. First we should check that these sequences really are arithmetic by taking diﬀerences of successive terms. Doing so will reveal the common diﬀerence d. 1. 5 − 2 3, 8 − 5 3, etc. To get from each term to the next, we add three, so d 3. The recursive deﬁnition is therefore a n a n−1 + 3 with a 0 2. The closed formula is a n 2 + 3n. 2. Here the common diﬀerence is −7, since we add −7 to 50 to get 43, and so on. Thus we have a recursive deﬁnition of a n a n−1 − 7 with a 0 50. The closed formula is a n 50 − 7n. What about sequences like 2, 6, 18, 54, . . .? This is not arithmetic because the diﬀerence between terms is not constant. However, the ratio between successive terms is constant. We call such sequences geometric. The recursive deﬁnition for the geometric sequence with initial term a and common ratio r is a n a n−1 · r; a 0 a. To get the next term we multiply the previous term by r. We can ﬁnd the closed formula like we did for the arithmetic progression. Write a0 a a1 a0 · r a2 a1 · r a0 · r · r a0 · r 2 150 2. Sequences .. . We must multiply the ﬁrst term a by r a number of times, n times to be precise. We get a n a · r n . Geometric Sequences. A sequence is called geometric if the ratio between successive terms is constant. Suppose the initial term a0 is a and the common ratio is r. Then we have, Recursive deﬁnition: a n ra n−1 with a 0 a. Closed formula: a n a · r n . Example 2.2.2 Find the recursive and closed formula for the geometric sequences below. Again, the ﬁrst term listed is a 0 . 1. 3, 6, 12, 24, 48, . . . 2. 27, 9, 3, 1, 1/3, . . . Solution. Start by checking that these sequences really are geometric by dividing each term by its previous term. If this ratio really is constant, we will have found r. 1. 6/3 2, 12/6 2, 24/12 2, etc. Yes, to get from any term to the next, we multiply by r 2. So the recursive deﬁnition is a n 2a n−1 with a 0 3. The closed formula is a n 3 · 2n . 2. The common ratio is r 1/3. So the sequence has recursive deﬁnition a n 31 a n−1 with a 0 27 and closed formula a n n 27 · 31 . In the examples and formulas above, we assumed that the initial term was a 0 . If your sequence starts with a 1 , you can easily ﬁnd the term that would have been a 0 and use that in the formula. For example, if we want a formula for the sequence 2, 5, 8, . . . and insist that 2 a 1 , then we can ﬁnd a 0 −1 (since the sequence is arithmetic with common diﬀerence 3, we have a 0 + 3 a 1 ). Then the closed formula will be a n −1 + 3n. Remark 2.2.3 If you look at other textbooks or online, you might ﬁnd that their closed formulas for arithmetic and geometric sequences diﬀer from ours. Speciﬁcally, you might ﬁnd the formulas a n a + (n − 1)d (arithmetic) and a n a · r n−1 (geometric). Which is correct? Both! In our case, we take a to be a 0 . If instead we had a 1 as our initial term, we would get the (slightly more complicated) formulas you ﬁnd elsewhere. 2.2. Arithmetic and Geometric Sequences 151 Sums of Arithmetic and Geometric Sequences Investigate! Your neighborhood grocery store has a candy machine full of Skittles. 1. Suppose that the candy machine currently holds exactly 650 Skittles, and every time someone inserts a quarter, exactly 7 Skittles come out of the machine. (a) How many Skittles will be left in the machine after 20 quarters have been inserted? (b) Will there ever be exactly zero Skittles left in the machine? Explain. 2. What if the candy machine gives 7 Skittles to the ﬁrst customer who put in a quarter, 10 to the second, 13 to the third, 16 to the fourth, etc. How many Skittles has the machine given out after 20 quarters are put into the machine? 3. Now, what if the machine gives 4 Skittles to the ﬁrst customer, 7 to the second, 12 to the third, 19 to the fourth, etc. How many Skittles has the machine given out after 20 quarters are put into the machine? ! Attempt the above activity before proceeding ! Look at the sequence (Tn )n≥1 which starts 1, 3, 6, 10, 15, . . .. These are called the triangular numbers since they represent the number of dots in an equilateral triangle (think of how you arrange 10 bowling pins: a row of 4 plus a row of 3 plus a row of 2 and a row of 1). T1 1 T2 3 T3 6 T4 10 Is this sequence arithmetic? No, since 3 − 1 2 and 6 − 3 3 , 2, so there is no common diﬀerence. Is the sequence geometric? No. 3/1 3 but 6/3 2, so there is no common ratio. What to do? Notice that the diﬀerences between terms do form an arithmetic sequence: 2, 3, 4, 5, 6, . . .. This means that the nth term of the sequence (Tn ) is the sum of the ﬁrst n terms in the sequence 1, 2, 3, 4, 5, . . .. We say that 152 2. Sequences (Tn ) is the sequence of partial sums of the sequence 1, 2, 3, . . . (partial sums because we are not taking the sum of all inﬁnitely many terms). This should become clearer if we write the triangular numbers like this: 11 31+2 61+2+3 10 1 + 2 + 3 + 4 .. .. . . Tn 1 + 2 + 3 + · · · + n. If we know how to add up the terms of an arithmetic sequence, we could ﬁnd a closed formula for a sequence whose diﬀerences are the terms of that arithmetic sequence. Consider how we could ﬁnd the sum of the ﬁrst 100 positive integers (that is, T100 ). Instead of adding them in order, we regroup and add 1 + 100 101. The next pair to combine is 2 + 99 101. Then 3 + 98 101. Keep going. This gives 50 pairs which each add up to 101, so T100 101 · 50 5050.1 n(n+1) In general, using this same sort of regrouping, we ﬁnd that Tn 2 . Incidentally, this is exactly the same as n+1 2 , which makes sense if you think of the triangular numbers as counting the number of handshakes that take place at a party with n + 1 people: the ﬁrst person shakes n hands, the next shakes an additional n − 1 hands and so on. The point of all of this is that some sequences, while not arithmetic or geometric, can be interpreted as the sequence of partial sums of arithmetic and geometric sequences. Luckily there are methods we can use to compute these sums quickly. Summing Arithmetic Sequences: Reverse and Add Here is a technique that allows us to quickly ﬁnd the sum of an arithmetic sequence. Example 2.2.4 Find the sum: 2 + 5 + 8 + 11 + 14 + · · · + 470. Solution. The idea is to mimic how we found the formula for triangular numbers. If we add the ﬁrst and last terms, we get 472. 1This insight is usually attributed to Carl Friedrich Gauss, one of the greatest mathematicians of all time, who discovered it as a child when his unpleasant elementary teacher thought he would keep the class busy by requiring them to compute the lengthy sum. 2.2. Arithmetic and Geometric Sequences 153 The second term and second-to-last term also add up to 472. To keep track of everything, we might express this as follows. Call the sum S. Then, S + S 2S 2 470 472 + + + 5 467 472 + + + 8 464 472 +···+ +···+ +···+ + + + 467 5 472 470 2 472 To ﬁnd 2S then we add 472 to itself a number of times. What number? We need to decide how many terms (summands) are in the sum. Since the terms form an arithmetic sequence, the nth term in the sum (counting 2 as the 0th term) can be expressed as 2 + 3n. If 2 + 3n 470 then n 156. So n ranges from 0 to 156, giving 157 terms in the sum. This is the number of 472’s in the sum for 2S. Thus 2S 157 · 472 74104. It is now easy to ﬁnd S: S 74104/2 37052. This will work for the sum of any arithmetic sequence. Call the sum S. Reverse and add. This produces a single number added to itself many times. Find the number of times. Multiply. Divide by 2. Done. Example 2.2.5 Find a closed formula for 6 + 10 + 14 + · · · + (4n − 2). Solution. Again, we have a sum of an arithmetic sequence. How many terms are in the sequence? Clearly each term in the sequence has the form 4k − 2 (as evidenced by the last term). For which values of k though? To get 6, k 2. To get 4n − 2 take k n. So to ﬁnd the number of terms, we must count the number of integers in the range 2, 3, . . . , n. This is n − 1. (There are n numbers from 1 to n, so one less if we start with 2.) Now reverse and add: + S S 2S 6 4n − 2 4n + 4 + + + 10 4n − 6 4n + 4 +···+ +···+ +···+ 4n − 6 10 4n + 4 + + + 4n − 2 6 4n + 4 Since there are n − 1 terms, we get 2S (n − 1)(4n + 4) so S (n − 1)(4n + 4) . 2 154 2. Sequences Besides ﬁnding sums, we can use this technique to ﬁnd closed formulas for sequences we recognize as sequences of partial sums. Example 2.2.6 Use partial sums to ﬁnd a closed formula for (a n )n≥0 which starts 2, 3, 7, 14, 24, 37, . . . . . . Solution. First, if you look at the diﬀerences between terms, you get a sequence of diﬀerences: 1, 4, 7, 10, 13, . . ., which is an arithmetic sequence. Written another way: a0 2 a1 2 + 1 a2 2 + 1 + 4 a3 2 + 1 + 4 + 7 and so on. We can write the general term of (a n ) in terms of the arithmetic sequence as follows: a n 2 + 1 + 4 + 7 + 10 + · · · + (1 + 3(n − 1)) (we use 1 + 3(n − 1) instead of 1 + 3n to get the indices to line up correctly; for a 3 we add up to 7, which is 1 + 3(3 − 1)). We can reverse and add, but the initial 2 does not ﬁt our pattern. This just means we need to keep the 2 out of the reverse part: + an an 2a n 2 2 4 + + + 1 1 + 3(n − 1) 2 + 3(n − 1) + + + 4 1 + 3(n − 2) 2 + 3(n − 1) +···+ +···+ +···+ 1 + 3(n − 1) 1 2 + 3(n − 1) Not counting the ﬁrst term (the 4) there are n summands of 2 + 3(n − 1) 3n − 1 so the right-hand side becomes 4 + (3n − 1)n. Finally, solving for a n we get an 4 + (3n − 1)n . 2 Just to be sure, we check a 0 correct closed formula. 4 2 2, a 1 4+2 2 3, etc. We have the Summing Geometric Sequences: Multiply, Shift and Subtract To ﬁnd the sum of a geometric sequence, we cannot just reverse and add. Do you see why? The reason we got the same term added to itself many times is because there was a constant diﬀerence. So as we added that diﬀerence in one direction, we subtracted the diﬀerence going the other 2.2. Arithmetic and Geometric Sequences 155 way, leaving a constant total. For geometric sums, we have a diﬀerent technique. Example 2.2.7 What is 3 + 6 + 12 + 24 + · · · + 12288? Solution. Multiply each term by 2, the common ratio. You get 2S 6 + 12 + 24 + · · · + 24576. Now subtract: 2S − S −3 + 24576 24573. Since 2S − S S, we have our answer. To better see what happened in the above example, try writing it this way: S 2S −S − 3+ 6 + 12 + 24 + · · · + 12288 6 + 12 + 24 + · · · + 12288 0+0+0+···+0 3+ +24576 −24576 Then divide both sides by −1 and we have the same result for S. The idea is, by multiplying the sum by the common ratio, each term becomes the next term. We shift over the sum to get the subtraction to mostly cancel out, leaving just the ﬁrst term and new last term. Example 2.2.8 Find a closed formula for S(n) 2 + 10 + 50 + · · · + 2 · 5n . Solution. The common ratio is 5. So we have − Thus S S 5S −4S 2 + 10 + 50 + · · · + 2 · 5n 10 + 50 + · · · + 2 · 5n + 2 · 5n+1 2 − 2 · 5n+1 2 − 2 · 5n+1 −4 Even though this might seem like a new technique, you have probably used it before. Example 2.2.9 Express 0.464646 . . . as a fraction. Solution. Let N 0.46464646 . . .. Consider 0.01N. We get: − N 0.01N 0.99N 0.4646464 . . . 0.00464646 . . . 0.46 156 2. Sequences So N 46 99 . What have we done? We viewed the repeating decimal 0.464646 . . . as a sum of the geometric sequence 0.46, 0.0046, 0.000046, . . . The common ratio is 0.01. The only real diﬀerence is that we are now computing an inﬁnite geometric sum, we do not have the extra “last” term to consider. Really, this is the result of taking a limit as you would in calculus when you compute inﬁnite geometric sums. Exercises 1. Consider the sequence 5, 9, 13, 17, 21, . . . with a 1 5 (a) Give a recursive deﬁnition for the sequence. (b) Give a closed formula for the nth term of the sequence. (c) Is 2013 a term in the sequence? Explain. (d) How many terms does the sequence 5, 9, 13, 17, 21, . . . , 533 have? (e) Find the sum: 5 + 9 + 13 + 17 + 21 + · · · + 533. Show your work. (f) Use what you found above to ﬁnd b n , the n th term of 1, 6, 15, 28, 45, . . ., where b0 1 2. Consider the sequence (a n )n≥0 which starts 8, 14, 20, 26, . . . . (a) What is the next term in the sequence? (b) Find a formula for the nth term of this sequence. (c) Find the sum of the ﬁrst 100 terms of the sequence: 3. Í99 k0 ak . Consider the sum 4 + 11 + 18 + 25 + · · · + 249. (a) How many terms (summands) are in the sum? (b) Compute the sum using a technique discussed in this section. 4. Consider the sequence 1, 7, 13, 19, . . . , 6n + 7. (a) How many terms are there in the sequence? Your answer will be in terms of n. (b) What is the second-to-last term? (c) Find the sum of all the terms in the sequence, in terms of n. 5. Find 5 + 7 + 9 + 11 + · · · + 521 using a technique from this section. 6. Find 5 + 15 + 45 + · · · + 5 · 320 . 7. Find 1 − 2 3 + 4 9 −···+ 230 . 330 2.2. Arithmetic and Geometric Sequences 157 8. Find x and y such that 27, x, y, 1 is part of an arithmetic sequence. Then ﬁnd x and y so that the sequence is part of a geometric sequence. (Warning: x and y might not be integers.) 9. Find x and y such that 5, x, y, 32 is part of an arithmetic sequence. Then ﬁnd x and y so that the sequence is part of a geometric sequence. (Warning: x and y might not be integers.) 10. Is there a pair of integers (a, b) such that a, x1 , y2 , b is part of an arithmetic sequences and a, x2 , y2 , b is part of a geometric sequence with x 1 , x 2 , y1 , y2 all integers? 11. Consider the sequence 2, 7, 15, 26, 40, 57, . . . (with a 0 2). By looking at the diﬀerences between terms, express the sequence as a sequence of partial sums. Then ﬁnd a closed formula for the sequence by computing the nth partial sum. 12. Starting with any rectangle, we can create a new, larger rectangle by attaching a square to the longer side. For example, if we start with a 2 × 5 rectangle, we would glue on a 5 × 5 square, forming a 5 × 7 rectangle: ⇝ 5 5 2 5 7 The next rectangle would be formed by attaching a 7 × 7 square to the top or bottom of the 5 × 7 rectangle. (a) Create a sequence of rectangles using this rule starting with a 1 × 2 rectangle. Then write out the sequence of perimeters for the rectangles (the ﬁrst term of the sequence would be 6, since the perimeter of a 1 × 2 rectangle is 6 - the next term would be 10). (b) Repeat the above part this time starting with a 1 × 3 rectangle. (c) Find recursive formulas for each of the sequences of perimeters you found in parts (a) and (b). Don’t forget to give the initial conditions as well. (d) Are the sequences arithmetic? Geometric? If not, are they close to being either of these (i.e., are the diﬀerences or ratios almost constant)? Explain. 158 2. Sequences 13. If you have enough toothpicks, you can make a large triangular grid. Below, are the triangular grids of size 1 and of size 2. The size 1 grid requires 3 toothpicks, the size 2 grid requires 9 toothpicks. (a) Let t n be the number of toothpicks required to make a size n triangular grid. Write out the ﬁrst 5 terms of the sequence t1 , t2 , . . .. (b) Find a recursive deﬁnition for the sequence. Explain why you are correct. (c) Is the sequence arithmetic or geometric? If not, is it the sequence of partial sums of an arithmetic or geometric sequence? Explain why your answer is correct. (d) Use your results from part (c) to ﬁnd a closed formula for the sequence. Show your work. 14. If you were to shade in a n × n square on graph paper, you could do it the boring way (with sides parallel to the edge of the paper) or the interesting way, as illustrated below: The interesting thing here, is that a 3 × 3 square now has area 13. Our goal is the ﬁnd a formula for the area of a n × n (diagonal) square. (a) Write out the ﬁrst few terms of the sequence of areas (assume a1 1, a 2 5, etc). Is the sequence arithmetic or geometric? If not, is it the sequence of partial sums of an arithmetic or geometric sequence? Explain why your answer is correct, referring to the diagonal squares. (b) Use your results from part (a) to ﬁnd a closed formula for the sequence. Show your work. Note, while there are lots of ways to ﬁnd a closed formula here, you should use partial sums specifically. 2.2. Arithmetic and Geometric Sequences 159 (c) Find the closed formula in as many other interesting ways as you can. 15. Here is a surprising use of sequences to answer a counting question: How many license plates consist of 6 symbols, using only the three numerals 1, 2, and 3 and the four letters a, b, c, and d, so that no numeral appears after any letter? For example, “31ddac” and “12321” are acceptable license plates, but “13ba2c” is not. (a) First answer this question by considering diﬀerent cases: how many of the license plates contain no numerals? How many contain one numeral, etc. (b) Now use the techniques of this section to show why the answer is 47 − 37 . 160 2. Sequences 2.3 Polynomial Fitting Investigate! A standard 8 × 8 chessboard contains 64 squares. Actually, this is just the number of unit squares. How many squares of all sizes are there on a chessboard? Start with smaller boards: 1 × 1, 2 × 2, 3 × 3, etc. Find a formula for the total number of squares in an n × n board. ! Attempt the above activity before proceeding ! So far we have seen methods for ﬁnding the closed formulas for arithmetic and geometric sequences. Since we know how to compute the sum of the ﬁrst n terms of arithmetic and geometric sequences, we can compute the closed formulas for sequences which have an arithmetic (or geometric) sequence of diﬀerences between terms. But what if we consider a sequence which is the sum of the ﬁrst n terms of a sequence which is itself the sum of an arithmetic sequence? Before we get too carried away, let’s consider an example: How many squares (of all sizes) are there on a chessboard? A chessboard consists of 64 squares, but we also want to consider squares of longer side length. Even though we are only considering an 8 × 8 board, there is already a lot to count. So instead, let us build a sequence: the ﬁrst term will be the number of squares on a 1 × 1 board, the second term will be the number of squares on a 2 × 2 board, and so on. After a little thought, we arrive at the sequence 1, 5, 14, 30, 55, . . . . This sequence is not arithmetic (or geometric for that matter), but perhaps it’s sequence of diﬀerences is. For diﬀerences we get 4, 9, 16, 25, . . . . Not a huge surprise: one way to count the number of squares in a 4 × 4 chessboard is to notice that there are 16 squares with side length 1, 9 with side length 2, 4 with side length 3 and 1 with side length 4. So the original sequence is just the sum of squares. Now this sequence of diﬀerences is not arithmetic since it’s sequence of diﬀerences (the diﬀerences of the diﬀerences of the original sequence) is not constant. In fact, this sequence of second diﬀerences is 5, 7, 9, . . . , which is an arithmetic sequence (with constant diﬀerence 2). Notice that our original sequence had third diﬀerences (that is, diﬀerences of diﬀerences of diﬀerences of the original) constant. We will call such a 2.3. Polynomial Fitting 161 sequence ∆3 -constant. The sequence 1, 4, 9, 16, . . . has second diﬀerences constant, so it will be a ∆2 -constant sequence. In general, we will say a sequence is a ∆k -constant sequence if the kth diﬀerences are constant. Example 2.3.1 Which of the following sequences are ∆k -constant for some value of k? 1. 2, 3, 7, 14, 24, 37, . . .. 2. 1, 8, 27, 64, 125, 216, . . .. 3. 1, 2, 4, 8, 16, 32, 64, . . .. Solution. 1. This is the sequence from Example 2.2.6, in which we found a closed formula by recognizing the sequence as the sequence of partial sums of an arithmetic sequence. Indeed, the sequence of ﬁrst diﬀerences is 1, 4, 7, 10, 13, . . ., which itself has diﬀerences 3, 3, 3, 3, . . .. Thus 2, 3, 7, 14, 24, 37, . . . is a ∆2 constant sequence. 2. These are the perfect cubes. The sequence of ﬁrst diﬀerences is 7, 19, 37, 61, 91, . . .; the sequence of second diﬀerences is 12, 18, 24, 30, . . .; the sequence of third diﬀerences is constant: 6, 6, 6, . . .. Thus the perfect cubes are a ∆3 -constant sequence. 3. If we take ﬁrst diﬀerences we get 1, 2, 4, 8, 16, . . .. Wait, what? That’s the sequence we started with. So taking second diﬀerences will give us the same sequence again. No matter how many times we repeat this we will always have the same sequence, which in particular means no ﬁnite number of diﬀerences will be constant. Thus this sequence is not ∆k -constant for any k. The ∆0 -constant sequences are themselves constant, so a closed formula for them is easy to compute (it’s just the constant). The ∆1 -constant sequences are arithmetic and we have a method for ﬁnding closed formulas for them as well. Every ∆2 -constant sequence is the sum of an arithmetic sequence so we can ﬁnd formulas for these as well. But notice that the format of the closed formula for a ∆2 -constant sequence is always quadratic. For example, the square numbers are ∆2 -constant with closed formula a n n 2 . The triangular numbers (also ∆2 -constant) have n(n+1) closed formula a n 2 , which when multiplied out gives you an n 2 162 2. Sequences term as well. It appears that every time we increase the complexity of the sequence, that is, increase the number of diﬀerences before we get constants, we also increase the degree of the polynomial used for the closed formula. We go from constant to linear to quadratic. The sequence of diﬀerences between terms tells us something about the rate of growth of the sequence. If a sequence is growing at a constant rate, then the formula for the sequence will be linear. If the sequence is growing at a rate which itself is growing at a constant rate, then the formula is quadratic. You have seen this elsewhere: if a function has a constant second derivative (rate of change) then the function must be quadratic. This works in general: Finite Diﬀerences. The closed formula for a sequence will be a degree k polynomial if and only if the sequence is ∆k -constant (i.e., the kth sequence of diﬀerences is constant). This tells us that the sequence of numbers of squares on a chessboard, 1, 5, 14, 30, 55, . . ., which we saw to be ∆3 -constant, will have a cubic (degree 3 polynomial) for its closed formula. Now once we know what format the closed formula for a sequence will take, it is much easier to actually ﬁnd the closed formula. In the case that the closed formula is a degree k polynomial, we just need k + 1 data points to “ﬁt” the polynomial to the data. Example 2.3.2 Find a formula for the sequence 3, 7, 14, 24, . . .. Assume a 1 3. Solution. First, check to see if the formula has constant diﬀerences at some level. The sequence of ﬁrst diﬀerences is 4, 7, 10, . . . which is arithmetic, so the sequence of second diﬀerences is constant. The sequence is ∆2 -constant, so the formula for a n will be a degree 2 polynomial. That is, we know that for some constants a, b, and c, a n an 2 + bn + c. Now to ﬁnd a, b, and c. First, it would be nice to know what a 0 is, since plugging in n 0 simpliﬁes the above formula greatly. In this case, a 0 2 (work backwards from the sequence of constant diﬀerences). Thus a 0 2 a · 02 + b · 0 + c, so c 2. Now plug in n 1 and n 2. We get a1 3 a + b + 2 2.3. Polynomial Fitting 163 a 2 7 a4 + b2 + 2. At this point we have two (linear) equations and two unknowns, so we can solve the system for a and b (using substitution or elimination or even matrices). We ﬁnd a 32 and b −1 2 , so a n 32 n 2 − 12 n + 2. Example 2.3.3 Find a closed formula for the number of squares on an n × n chessboard. Solution. We have seen that the sequence 1, 5, 14, 30, 55, . . . is ∆3 constant, so we are looking for a degree 3 polynomial. That is, a n an 3 + bn 2 + cn + d. We can ﬁnd d if we know what a 0 is. Working backwards from the third diﬀerences, we ﬁnd a 0 0 (unsurprisingly, since there are no squares on a 0 × 0 chessboard). Thus d 0. Now plug in n 1, n 2, and n 3: 1 a + b + c 5 8a + 4b + 2c 14 27a + 9b + 3c. If we solve this system of equations we get a 13 , b 21 and c 61 . Therefore the number of squares on an n × n chessboard is a n 31 n 3 + 12 n 2 + 16 n 16 n(n + 1)(2n + 1). Note: Since the squares-on-a-chessboard problem is really asking for the sum of squares, we now have a nice formula for n Õ k2. k1 Not all sequences will have polynomials as their closed formula. We can use the theory of ﬁnite diﬀerences to identify these. Example 2.3.4 Determine whether the following sequences can be described by a polynomial, and if so, of what degree. 1. 1, 2, 4, 8, 16, . . . 2. 0, 7, 50, 183, 484, 1055, . . . 164 2. Sequences 3. 1, 1, 2, 3, 5, 8, 13, . . . Solution. 1. As we saw in Example 2.3.1, this sequence is not ∆k -constant for any k. Therefore the closed formula for the sequence is not a polynomial. In fact, we know the closed formula is a n 2n , which grows faster than any polynomial (so is not a polynomial). 2. The sequence of ﬁrst diﬀerences is 7, 43, 133, 301, 571, . . .. The second diﬀerences are: 36, 90, 168, 270, . . .. Third diﬀerence: 54, 78, 102, . . .. Fourth diﬀerences: 24, 24, . . .. As far as we can tell, this sequence of diﬀerences is constant so the sequence is ∆4 -constant and as such the closed formula is a degree 4 polynomial. 3. This is the Fibonacci sequence. The sequence of ﬁrst diﬀerences is 0, 1, 1, 2, 3, 5, 8, . . ., the sequence of second diﬀerences are 1, 0, 1, 1, 2, 3, 5 . . .. We notice that after the ﬁrst few terms, we get the original sequence back. So there will never be constant diﬀerences, so the closed formula for the Fibonacci sequence is not a polynomial. Exercises 1. Use polynomial ﬁtting to ﬁnd the formula for the nth term of the sequence (a n )n≥0 which starts, 0, 2, 6, 12, 20, . . . . 2. Use polynomial ﬁtting to ﬁnd the formula for the nth term of the sequence (a n )n≥0 which starts, 1, 2, 4, 8, 15, 26 . . . . 3. Use polynomial ﬁtting to ﬁnd the formula for the nth term of the sequence (a n )n≥0 which starts, 2, 5, 11, 21, 36, . . . . 4. Use polynomial ﬁtting to ﬁnd the formula for the nth term of the sequence (a n )n≥0 which starts, 3, 6, 12, 22, 37, 58, . . . . 2.3. Polynomial Fitting 5. 165 Make up sequences that have (a) 3, 3, 3, 3, . . . as its second diﬀerences. (b) 1, 2, 3, 4, 5, . . . as its third diﬀerences. (c) 1, 2, 4, 8, 16, . . . as its 100th diﬀerences. 6. Consider the sequence 1, 3, 7, 13, 21, . . .. Explain how you know the closed formula for the sequence will be quadratic. Then “guess” the correct formula by comparing this sequence to the squares 1, 4, 9, 16, . . . (do not use polynomial ﬁtting). 7. Use a similar technique as in the previous exercise to ﬁnd a closed formula for the sequence 2, 11, 34, 77, 146, 247, . . .. 8. Suppose a n n 2 + 3n + 4. Find a closed formula for the sequence of diﬀerences by computing a n − a n−1 . 9. Generalize Exercise 2.3.8: Find a closed formula for the sequence of diﬀerences of a n an 2 + bn + c. That is, prove that every quadratic sequence has arithmetic diﬀerences. 10. Can you use polynomial ﬁtting to ﬁnd the formula for the nth term of the sequence 4, 7, 11, 18, 29, 47, . . . ? Explain why or why not. 11. Will the nth sequence of diﬀerences of 2, 6, 18, 54, 162, . . . ever be constant? Explain. 12. In their down time, ghost pirates enjoy stacking cannonballs in triangular based pyramids (aka, tetrahedrons), like those pictured here: Note, these are solid tetrahedrons, so there will be some cannonballs obscured from view (the picture on the right has one cannonball in the back not shown in the picture, for example) The pirates wonder how many cannonballs would be required to build a pyramid 15 layers high (thus breaking the world cannonball stacking record). Can you help? (a) Let P(n) denote the number of cannonballs needed to create a pyramid n layers high. So P(1) 1, P(2) 4, and so on. Calculate P(3), P(4) and P(5). (b) Use polynomial ﬁtting to ﬁnd a closed formula for P(n). Show your work. 166 2. Sequences (c) Answer the pirate’s question: how many cannonballs do they need to make a pyramid 15 layers high? (d) Bonus: Locate this sequence in Pascal’s triangle. Why does that make sense? 2.4. Solving Recurrence Relations 2.4 167 Solving Recurrence Relations Investigate! Consider the recurrence relation a n 5a n−1 − 6a n−2 . 1. What sequence do you get if the initial conditions are a 0 1, a 1 2? Give a closed formula for this sequence. 2. What sequence do you get if the initial conditions are a 0 1, a 1 3? Give a closed formula. 3. What if a 0 2 and a 1 5? Find a closed formula. ! Attempt the above activity before proceeding ! We have seen that it is often easier to ﬁnd recursive deﬁnitions than closed formulas. Lucky for us, there are a few techniques for converting recursive deﬁnitions to closed formulas. Doing so is called solving a recurrence relation. Recall that the recurrence relation is a recursive deﬁnition without the initial conditions. For example, the recurrence relation for the Fibonacci sequence is Fn Fn−1 + Fn−2 . (This, together with the initial conditions F0 0 and F1 1 give the entire recursive deﬁnition for the sequence.) Example 2.4.1 Find a recurrence relation and initial conditions for 1, 5, 17, 53, 161, 485 . . .. Solution. Finding the recurrence relation would be easier if we had some context for the problem (like the Tower of Hanoi, for example). Alas, we have only the sequence. Remember, the recurrence relation tells you how to get from previous terms to future terms. What is going on here? We could look at the diﬀerences between terms: 4, 12, 36, 108, . . .. Notice that these are growing by a factor of 3. Is the original sequence as well? 1 · 3 3, 5 · 3 15, 17 · 3 51 and so on. It appears that we always end up with 2 less than the next term. Aha! So a n 3a n−1 + 2 is our recurrence relation and the initial condition is a 0 1. We are going to try to solve these recurrence relations. By this we mean something very similar to solving diﬀerential equations: we want to ﬁnd a function of n (a closed formula) which satisﬁes the recurrence relation, 168 2. Sequences as well as the initial condition.2 Just like for diﬀerential equations, ﬁnding a solution might be tricky, but checking that the solution is correct is easy. Example 2.4.2 Check that a n 2n + 1 is a solution to the recurrence relation a n 2a n−1 − 1 with a 1 3. Solution. First, it is easy to check the initial condition: a 1 should be 21 + 1 according to our closed formula. Indeed, 21 + 1 3, which is what we want. To check that our proposed solution satisﬁes the recurrence relation, try plugging it in. 2a n−1 − 1 2(2n−1 + 1) − 1 2n + 2 − 1 2n + 1 an . That’s what our recurrence relation says! We have a solution. Sometimes we can be clever and solve a recurrence relation by inspection. We generate the sequence using the recurrence relation and keep track of what we are doing so that we can see how to jump to ﬁnding just the a n term. Here are two examples of how you might do that. Telescoping refers to the phenomenon when many terms in a large sum cancel out - so the sum “telescopes.” For example: (2 − 1) + (3 − 2) + (4 − 3) + · · · + (100 − 99) + (101 − 100) −1 + 101 because every third term looks like: 2 + −2 0, and then 3 + −3 0 and so on. We can use this behavior to solve recurrence relations. Here is an example. Example 2.4.3 Solve the recurrence relation a n a n−1 + n with initial term a 0 4. Solution. To get a feel for the recurrence relation, write out the ﬁrst few terms of the sequence: 4, 5, 7, 10, 14, 19, . . .. Look at the diﬀerence between terms. a 1 − a 0 1 and a 2 − a 1 2 and so on. The key thing here is that the diﬀerence between terms is n. We can write this explicitly: a n − a n−1 n. Of course, we could have 2Recurrence relations are sometimes called diﬀerence equations since they can describe the diﬀerence between terms and this highlights the relation to diﬀerential equations further. 2.4. Solving Recurrence Relations 169 arrived at this conclusion directly from the recurrence relation by subtracting a n−1 from both sides. Now use this equation over and over again, changing n each time: a1 − a0 1 a2 − a1 2 a3 − a2 3 .. .. . . a n − a n−1 n. Add all these equations together. On the right-hand side, we get the sum 1+2+3+· · ·+n. We already know this can be simpliﬁed n(n+1) to 2 . What happens on the left-hand side? We get (a 1 − a 0 ) + (a 2 − a 1 ) + (a 3 − a 2 ) + · · · (a n−1 − a n−2 ) + (a n − a n−1 ). This sum telescopes. We are left with only the −a 0 from the ﬁrst equation and the a n from the last equation. Putting this all n(n+1) n(n+1) together we have −a 0 + a n 2 or a n 2 + a 0 . But we know that a 0 4. So the solution to the recurrence relation, subject to the initial condition is n(n + 1) an + 4. 2 (Now that we know that, we should notice that the sequence is the result of adding 4 to each of the triangular numbers.) The above example shows a way to solve recurrence relations of the Í form a n a n−1 + f (n) where nk1 f (k) has a known closed formula. If you rewrite the recurrence relation as a n − a n−1 f (n), and then add up all the diﬀerent equations with n ranging between 1 and n, the left-hand Í side will always give you a n − a 0 . The right-hand side will be nk1 f (k), which is why we need to know the closed formula for that sum. However, telescoping will not help us with a recursion such as a n 3a n−1 +2 since the left-hand side will not telescope. You will have −3a n−1 ’s but only one a n−1 . However, we can still be clever if we use iteration. We have already seen an example of iteration when we found the closed formula for arithmetic and geometric sequences. The idea is, we iterate the process of ﬁnding the next term, starting with the known initial condition, up until we have a n . Then we simplify. In the arithmetic sequence example, we simpliﬁed by multiplying d by the number of times 170 2. Sequences we add it to a when we get to a n , to get from a n a + d + d + d + · · · + d to a n a + dn. To see how this works, let’s go through the same example we used for telescoping, but this time use iteration. Example 2.4.4 Use iteration to solve the recurrence relation a n a n−1 + n with a 0 4. Solution. Again, start by writing down the recurrence relation when n 1. This time, don’t subtract the a n−1 terms to the other side: a 1 a 0 + 1. Now a 2 a 1 + 2, but we know what a 1 is. By substitution, we get a 2 (a 0 + 1) + 2. Now go to a3 a 2 + 3, using our known value of a 2 : a3 ((a 0 + 1) + 2) + 3. We notice a pattern. Each time, we take the previous term and add the current index. So a n ((((a 0 + 1) + 2) + 3) + · · · + n − 1) + n. Regrouping terms, we notice that a n is just a 0 plus the sum of the integers from 1 to n. So, since a 0 4, an 4 + n(n + 1) . 2 Of course in this case we still needed to know formula for the sum of 1, . . . , n. Let’s try iteration with a sequence for which telescoping doesn’t work. Example 2.4.5 Solve the recurrence relation a n 3a n−1 + 2 subject to a 0 1. Solution. Again, we iterate the recurrence relation, building up to the index n. a 1 3a 0 + 2 a 2 3(a 1 ) + 2 3(3a 0 + 2) + 2 2.4. Solving Recurrence Relations 171 a 3 3[a 2 ] + 2 3[3(3a 0 + 2) + 2] + 2 .. .. .. . . . a n 3(a n−1 ) + 2 3(3(3(3 · · · (3a 0 + 2) + 2) + 2) · · · + 2) + 2. It is diﬃcult to see what is happening here because we have to distribute all those 3’s. Let’s try again, this time simplifying a bit as we go. a 1 3a 0 + 2 a 2 3(a 1 ) + 2 3(3a 0 + 2) + 2 32 a 0 + 2 · 3 + 2 a 3 3[a 2 ] + 2 3[32 a 0 + 2 · 3 + 2] + 2 33 a 0 + 2 · 32 + 2 · 3 + 2 .. .. .. . . . a n 3(a n−1 ) + 2 3(3n−1 a 0 + 2 · 3n−2 + · · · + 2) + 2 3n a 0 + 2 · 3n−1 + 2 · 3n−2 + · · · + 2 · 3 + 2. Now we simplify. a 0 1, so we have 3n + ⟨stuﬀ⟩. Note that all the other terms have a 2 in them. In fact, we have a geometric sum with ﬁrst term 2 and common ratio 3. We have seen how to simplify n 2 + 2 · 3 + 2 · 32 + · · · + 2 · 3n−1 . We get 2−2·3 which simpliﬁes to −2 3n − 1. Putting this together with the ﬁrst 3n term gives our closed formula: a n 2 · 3n − 1. Iteration can be messy, but when the recurrence relation only refers to one previous term (and maybe some function of n) it can work well. However, trying to iterate a recurrence relation such as a n 2a n−1 + 3a n−2 will be way too complicated. We would need to keep track of two sets of previous terms, each of which were expressed by two previous terms, and so on. The length of the formula would grow exponentially (double each time, in fact). Luckily there happens to be a method for solving recurrence relations which works very well on relations like this. The Characteristic Root Technique Suppose we want to solve a recurrence relation expressed as a combination of the two previous terms, such as a n a n−1 + 6a n−2 . In other words, we want to ﬁnd a function of n which satisﬁes a n − a n−1 − 6a n−2 0. Now iteration is too complicated, but think just for a second what would happen if we did iterate. In each step, we would, among other things, multiply a previous iteration by 6. So our closed formula would include 6 multiplied 172 2. Sequences some number of times. Thus it is reasonable to guess the solution will contain parts that look geometric. Perhaps the solution will take the form r n for some constant r. The nice thing is, we know how to check whether a formula is actually a solution to a recurrence relation: plug it in. What happens if we plug in r n into the recursion above? We get r n − r n−1 − 6r n−2 0. Now solve for r: r n−2 (r 2 − r − 6) 0, so by factoring, r −2 or r 3 (or r 0, although this does not help us). This tells us that a n (−2)n is a solution to the recurrence relation, as is a n 3n . Which one is correct? They both are, unless we specify initial conditions. Notice we could also have a n (−2)n + 3n . Or a n 7(−2)n + 4 · 3n . In fact, for any a and b, a n a(−2)n + b3n is a solution (try plugging this into the recurrence relation). To ﬁnd the values of a and b, use the initial conditions. This points us in the direction of a more general technique for solving recurrence relations. Notice we will always be able to factor out the r n−2 as we did above. So we really only care about the other part. We call this other part the characteristic equation for the recurrence relation. We are interested in ﬁnding the roots of the characteristic equation, which are called (surprise) the characteristic roots. Characteristic Roots. Given a recurrence relation a n +αa n−1 +βa n−2 0, the characteristic polynomial is x 2 + αx + β giving the characteristic equation: x 2 + αx + β 0. If r1 and r2 are two distinct roots of the characteristic polynomial (i.e, solutions to the characteristic equation), then the solution to the recurrence relation is a n ar1n + br2n , where a and b are constants determined by the initial conditions. 2.4. Solving Recurrence Relations 173 Example 2.4.6 Solve the recurrence relation a n 7a n−1 − 10a n−2 with a 0 2 and a 1 3. Solution. Rewrite the recurrence relation a n − 7a n−1 + 10a n−2 0. Now form the characteristic equation: x 2 − 7x + 10 0 and solve for x: (x − 2)(x − 5) 0 so x 2 and x 5 are the characteristic roots. We therefore know that the solution to the recurrence relation will have the form a n a2n + b5n . To ﬁnd a and b, plug in n 0 and n 1 to get a system of two equations with two unknowns: 2 a20 + b50 a + b 3 a21 + b51 2a + 5b Solving this system gives a the recurrence relation is 7 3 and b − 13 so the solution to 7 1 a n 2n − 5n . 3 3 Perhaps the most famous recurrence relation is Fn Fn−1 +Fn−2 , which together with the initial conditions F0 0 and F1 1 deﬁnes the Fibonacci sequence. But notice that this is precisely the type of recurrence relation on which we can use the characteristic root technique. When you do, the only thing that changes is that the characteristic equation does not factor, so you need to use the quadratic formula to ﬁnd the characteristic roots. In fact, doing so gives the third most famous irrational number, φ, the golden ratio. Before leaving the characteristic root technique, we should think about what might happen when you solve the characteristic equation. We have an example above in which the characteristic polynomial has two distinct roots. These roots can be integers, or perhaps irrational numbers (requiring the quadratic formula to ﬁnd them). In these cases, we know what the solution to the recurrence relation looks like. However, it is possible for the characteristic polynomial to have only one root. This can happen if the characteristic polynomial factors as 174 2. Sequences (x − r)2 . It is still the case that r n would be a solution to the recurrence relation, but we won’t be able to ﬁnd solutions for all initial conditions using the general form a n ar1n + br2n , since we can’t distinguish between r1n and r2n . We are in luck though: Characteristic Root Technique for Repeated Roots. Suppose the recurrence relation a n αa n−1 + βa n−2 has a characteristic polynomial with only one root r. Then the solution to the recurrence relation is a n ar n + bnr n where a and b are constants determined by the initial conditions. Notice the extra n in bnr n . This allows us to solve for the constants a and b from the initial conditions. Example 2.4.7 Solve the recurrence relation a n 6a n−1 − 9a n−2 with initial conditions a 0 1 and a 1 4. Solution. The characteristic polynomial is x 2 − 6x + 9. We solve the characteristic equation x 2 − 6x + 9 0 by factoring: (x − 3)2 0 so x 3 is the only characteristic root. Therefore we know that the solution to the recurrence relation has the form a n a3n + bn3n for some constants a and b. Now use the initial conditions: a 0 1 a30 + b · 0 · 30 a a 1 4 a · 3 + b · 1 · 3 3a + 3b. Since a 1, we ﬁnd that b recurrence relation is 1 3. Therefore the solution to the 1 a n 3n + n3n . 3 2.4. Solving Recurrence Relations 175 Although we will not consider examples more complicated than these, this characteristic root technique can be applied to much more complicated recurrence relations. For example, a n 2a n−1 + a n−2 − 3a n−3 has characteristic polynomial x 3 − 2x 2 − x + 3. Assuming you see how to factor such a degree 3 (or more) polynomial you can easily ﬁnd the characteristic roots and as such solve the recurrence relation (the solution would look like a n ar1n + br2n + cr3n if there were 3 distinct roots). It is also possible that the characteristics roots are complex numbers. However, the characteristic root technique is only useful for solving recurrence relations in a particular form: a n is given as a linear combination of some number of previous terms. These recurrence relations are called linear homogeneous recurrence relations with constant coefﬁcients. The “homogeneous” refers to the fact that there is no additional term in the recurrence relation other than a multiple of a j terms. For example, a n 2a n−1 + 1 is non-homogeneous because of the additional constant 1. There are general methods of solving such things, but we will not consider them here, other than through the use of telescoping or iteration described above. Exercises 1. Find the next two terms in (a n )n≥0 beginning 3, 5, 11, 21, 43, 85 . . .. Then give a recursive deﬁnition for the sequence. Finally, use the characteristic root technique to ﬁnd a closed formula for the sequence. 2. Consider the sequences 2, 5, 12, 29, 70, 169, 408, . . . (with a 0 2). (a) Describe the rate of growth of this sequence. (b) Find a recursive deﬁnition for the sequence. (c) Find a closed formula for the sequence. (d) If you look at the sequence of diﬀerences between terms, and then the sequence of second diﬀerences, the sequence of third diﬀerences, and so on, will you ever get a constant sequence? Explain how you know. 3. Solve the recurrence relation a n a n−1 + 2n with a 0 5. 4. Show that 4n is a solution to the recurrence relation a n 3a n−1 +4a n−2 . 5. Find the solution to the recurrence relation a n 3a n−1 + 4a n−2 with initial terms a 0 2 and a 1 3. 6. Find the solution to the recurrence relation a n 3a n−1 + 4a n−2 with initial terms a 0 5 and a 1 8. 176 2. Sequences 7. Solve the recurrence relation a n 3a n−1 + 10a n−2 with initial terms a 0 4 and a 1 1. 8. Suppose that r n and q n are both solutions to a recurrence relation of the form a n αa n−1 + βa n−2 . Prove that c · r n + d · q n is also a solution to the recurrence relation, for any constants c, d. 9. Think back to the magical candy machine at your neighborhood grocery store. Suppose that the ﬁrst time a quarter is put into the machine 1 Skittle comes out. The second time, 4 Skittles, the third time 16 Skittles, the fourth time 64 Skittles, etc. (a) Find both a recursive and closed formula for how many Skittles the nth customer gets. (b) Check your solution for the closed formula by solving the recurrence relation using the Characteristic Root technique. 10. Let a n be the number of 1 × n tile designs you can make using 1 × 1 squares available in 4 colors and 1 × 2 dominoes available in 5 colors. (a) First, ﬁnd a recurrence relation to describe the problem. Explain why the recurrence relation is correct (in the context of the problem). (b) Write out the ﬁrst 6 terms of the sequence a 1 , a 2 , . . .. (c) Solve the recurrence relation. That is, ﬁnd a closed formula. 11. You have access to 1 × 1 tiles which come in 2 diﬀerent colors and 1 × 2 tiles which come in 3 diﬀerent colors. We want to ﬁgure out how many diﬀerent 1 × n path designs we can make out of these tiles. (a) Find a recursive deﬁnition for the number of paths of length n. (b) Solve the recurrence relation using the Characteristic Root technique. 12. Solve the recurrence relation a n 2a n−1 − a n−2 . (a) What is the solution if the initial terms are a 0 1 and a 1 2? (b) What do the initial terms need to be in order for a9 30? (c) For which x are there initial terms which make a 9 x? 13. Consider the recurrence relation a n 4a n−1 − 4a n−2 . (a) Find the general solution to the recurrence relation (beware the repeated root). (b) Find the solution when a 0 1 and a 1 2. (c) Find the solution when a 0 1 and a 1 8. 2.5. Induction 2.5 177 Induction Mathematical induction is a proof technique, not unlike direct proof or proof by contradiction or combinatorial proof.3 In other words, induction is a style of argument we use to convince ourselves and others that a mathematical statement is always true. Many mathematical statements can be proved by simply explaining what they mean. Others are very diﬃcult to prove—in fact, there are relatively simple mathematical statements which nobody yet knows how to prove. To facilitate the discovery of proofs, it is important to be familiar with some standard styles of arguments. Induction is one such style. Let’s start with an example: Stamps Investigate! You need to mail a package, but don’t yet know how much postage you will need. You have a large supply of 8-cent stamps and 5cent stamps. Which amounts of postage can you make exactly using these stamps? Which amounts are impossible to make? ! Attempt the above activity before proceeding ! Perhaps in investigating the problem above you picked some amounts of postage, and then ﬁgured out whether you could make that amount using just 8-cent and 5-cent stamps. Perhaps you did this in order: can you make 1 cent of postage? Can you make 2 cents? 3 cents? And so on. If this is what you did, you were actually answering a sequence of questions. We have methods for dealing with sequences. Let’s see if that helps. Actually, we will not make a sequence of questions, but rather a sequence of statements. Let P(n) be the statement “you can make n cents of postage using just 8-cent and 5-cent stamps.” Since for each value of n, P(n) is a statement, it is either true or false. So if we form the sequence of statements P(1), P(2), P(3), P(4), . . . , the sequence will consist of T’s (for true) and F’s (for false). In our particular case the sequence starts F, F, F, F, T, F, F, T, F, T, F, F, T, . . . because P(1), P(2), P(3), P(4) are all false (you cannot make 1, 2, 3, or 4 cents of postage) but P(5) is true (use one 5-cent stamp), and so on. Let’s think a bit about how we could ﬁnd the value of P(n) for some speciﬁc n (the “value” will be either T or F). How did we ﬁnd the value of 3You might or might not be familiar with these yet. We will consider these in Chapter 3. 178 2. Sequences the nth term of a sequence of numbers? How did we ﬁnd a n ? There were two ways we could do this: either there was a closed formula for a n , so we could plug in n into the formula and get our output value, or we had a recursive deﬁnition for the sequence, so we could use the previous terms of the sequence to compute the nth term. When dealing with sequences of statements, we could use either of these techniques as well. Maybe there is a way to use n itself to determine whether we can make n cents of postage. That would be something like a closed formula. Or instead we could use the previous terms in the sequence (of statements) to determine whether we can make n cents of postage. That is, if we know the value of P(n − 1), can we get from that to the value of P(n)? That would be something like a recursive deﬁnition for the sequence. Remember, ﬁnding recursive deﬁnitions for sequences was often easier than ﬁnding closed formulas. The same is true here. Suppose I told you that P(43) was true (it is). Can you determine from this fact the value of P(44) (whether it true or false)? Yes you can. Even if we don’t know how exactly we made 43 cents out of the 5-cent and 8-cent stamps, we do know that there was some way to do it. What if that way used at least three 5-cent stamps (making 15 cents)? We could replace those three 5-cent stamps with two 8-cent stamps (making 16 cents). The total postage has gone up by 1, so we have a way to make 44 cents, so P(44) is true. Of course, we assumed that we had at least three 5-cent stamps. What if we didn’t? Then we must have at least three 8-cent stamps (making 24 cents). If we replace those three 8-cent stamps with ﬁve 5-cent stamps (making 25 cents) then again we have bumped up our total by 1 cent so we can make 44 cents, so P(44) is true. Notice that we have not said how to make 44 cents, just that we can, on the basis that we can make 43 cents. How do we know we can make 43 cents? Perhaps because we know we can make 42 cents, which we know we can do because we know we can make 41 cents, and so on. It’s a recursion! As with a recursive deﬁnition of a numerical sequence, we must specify our initial value. In this case, the initial value is “P(1) is false.” That’s not good, since our recurrence relation just says that P(k + 1) is true if P(k) is also true. We need to start the process with a true P(k). So instead, we might want to use “P(28) is true” as the initial condition. Putting this all together we arrive at the following fact: it is possible to (exactly) make any amount of postage greater than 27 cents using just 5-cent and 8-cent stamps.4 In other words, P(k) is true for any k ≥ 28. To prove this, we could do the following: 1. Demonstrate that P(28) is true. 4This is not claiming that there are no amounts less than 27 cents which can also be made. 2.5. Induction 179 2. Prove that if P(k) is true, then P(k + 1) is true (for any k ≥ 28). Suppose we have done this. Then we know that the 28th term of the sequence above is a T (using step 1, the initial condition or base case), and that every term after the 28th is T also (using step 2, the recursive part or inductive case). Here is what the proof would actually look like. Proof. Let P(n) be the statement “it is possible to make exactly n cents of postage using 5-cent and 8-cent stamps.” We will show P(n) is true for all n ≥ 28. First, we show that P(28) is true: 28 4 · 5 + 1 · 8, so we can make 28 cents using four 5-cent stamps and one 8-cent stamp. Now suppose P(k) is true for some arbitrary k ≥ 28. Then it is possible to make k cents using 5-cent and 8-cent stamps. Note that since k ≥ 28, it cannot be that we use fewer than three 5-cent stamps and fewer than three 8-cent stamps: using two of each would give only 26 cents. Now if we have made k cents using at least three 5-cent stamps, replace three 5-cent stamps by two 8-cent stamps. This replaces 15 cents of postage with 16 cents, moving from a total of k cents to k + 1 cents. Thus P(k + 1) is true. On the other hand, if we have made k cents using at least three 8-cent stamps, then we can replace three 8-cent stamps with ﬁve 5-cent stamps, moving from 24 cents to 25 cents, giving a total of k + 1 cents of postage. So in this case as well P(k + 1) is true. Therefore, by the principle of mathematical induction, P(n) is true for all n ≥ 28. qed Formalizing Proofs What we did in the stamp example above works for many types of problems. Proof by induction is useful when trying to prove statements about all natural numbers, or all natural numbers greater than some ﬁxed ﬁrst case (like 28 in the example above), and in some other situations too. In particular, induction should be used when there is some way to go from one case to the next – when you can see how to always “do one more.” This is a big idea. Thinking about a problem inductively can give new insight into the problem. For example, to really understand the stamp problem, you should think about how any amount of postage (greater than 28 cents) can be made (this is non-inductive reasoning) and also how the ways in which postage can be made changes as the amount increases (inductive reasoning). When you are asked to provide a proof by induction, you are being asked to think about the problem dynamically; how does increasing n change the problem? But there is another side to proofs by induction as well. In mathematics, it is not enough to understand a problem, you must also be able to communicate the problem to others. Like any discipline, mathematics 180 2. Sequences has standard language and style, allowing mathematicians to share their ideas eﬃciently. Proofs by induction have a certain formal style, and being able to write in this style is important. It allows us to keep our ideas organized and might even help us with formulating a proof. Here is the general structure of a proof by mathematical induction: Induction Proof Structure. Start by saying what the statement is that you want to prove: “Let P(n) be the statement. . . ” To prove that P(n) is true for all n ≥ 0, you must prove two facts: 1. Base case: Prove that P(0) is true. You do this directly. This is often easy. 2. Inductive case: Prove that P(k) → P(k + 1) for all k ≥ 0. That is, prove that for any k ≥ 0 if P(k) is true, then P(k + 1) is true as well. This is the proof of an if . . . then . . . statement, so you can assume P(k) is true (P(k) is called the inductive hypothesis). You must then explain why P(k + 1) is also true, given that assumption. Assuming you are successful on both parts above, you can conclude, “Therefore by the principle of mathematical induction, the statement P(n) is true for all n ≥ 0.” Sometimes the statement P(n) will only be true for values of n ≥ 4, for example, or some other value. In such cases, replace all the 0’s above with 4’s (or the other value). The other advantage of formalizing inductive proofs is it allows us to verify that the logic behind this style of argument is valid. Why does induction work? Think of a row of dominoes set up standing on their edges. We want to argue that in a minute, all the dominoes will have fallen down. For this to happen, you will need to push the ﬁrst domino. That is the base case. It will also have to be that the dominoes are close enough together that when any particular domino falls, it will cause the next domino to fall. That is the inductive case. If both of these conditions are met, you push the ﬁrst domino over and each domino will cause the next to fall, then all the dominoes will fall. Induction is powerful! Think how much easier it is to knock over dominoes when you don’t have to push over each domino yourself. You just start the chain reaction, and the rely on the relative nearness of the dominoes to take care of the rest. Think about our study of sequences. It is easier to ﬁnd recursive deﬁnitions for sequences than closed formulas. Going from one case to the next is easier than going directly to a particular case. That is what is 2.5. Induction 181 so great about induction. Instead of going directly to the (arbitrary) case for n, we just need to say how to get from one case to the next. When you are asked to prove a statement by mathematical induction, you should ﬁrst think about why the statement is true, using inductive reasoning. Explain why induction is the right thing to do, and roughly why the inductive case will work. Then, sit down and write out a careful, formal proof using the structure above. Examples Here are some examples of proof by mathematical induction. Example 2.5.1 Prove for each natural number n ≥ 1 that 1 + 2 + 3 + · · · + n n(n+1) 2 . Solution. First, let’s think inductively about this equation. In fact, we know this is true for other reasons (reverse and add comes to mind). But why might induction be applicable? The left-hand side adds up the numbers from 1 to n. If we know how to do that, adding just one more term (n + 1) would not be that hard. For example, if n 100, suppose we know that the sum of the ﬁrst 100 numbers is 5050 (so 1 + 2 + 3 + · · · + 100 5050, which is true). Now to ﬁnd the sum of the ﬁrst 101 numbers, it makes more sense to just add 101 to 5050, instead of computing the entire sum again. We would have 1 + 2 + 3 + · · · + 100 + 101 5050 + 101 5151. In fact, it would always be easy to add just one more term. This is why we should use induction. Now the formal proof: n(n+1) Proof. Let P(n) be the statement 1 + 2 + 3 + · · · + n 2 . We will show that P(n) is true for all natural numbers n ≥ 1. 1(1+1) Base case: P(1) is the statement 1 2 which is clearly true. Inductive case: Let k ≥ 1 be a natural number. Assume (for k(k+1) induction) that P(k) is true. That means 1 + 2 + 3 + · · · + k 2 . We will prove that P(k + 1) is true as well. That is, we must prove (k+1)(k+2) that 1 + 2 + 3 + · · · + k + (k + 1) . To prove this equation, 2 start by adding k + 1 to both sides of the inductive hypothesis: 1 + 2 + 3 + · · · + k + (k + 1) k(k + 1) + (k + 1). 2 Now, simplifying the right side we get: k(k + 1) k(k + 1) 2(k + 1) +k+1 + 2 2 2 182 2. Sequences k(k + 1) + 2(k + 1) 2 (k + 2)(k + 1) . 2 Thus P(k + 1) is true, so by the principle of mathematical induction P(n) is true for all natural numbers n ≥ 1. qed Note that in the part of the proof in which we proved P(k+1) from P(k), we used the equation P(k). This was the inductive hypothesis. Seeing how to use the inductive hypotheses is usually straight forward when proving a fact about a sum like this. In other proofs, it can be less obvious where it ﬁts in. Example 2.5.2 Prove that for all n ∈ N, 6n − 1 is a multiple of 5. Solution. Again, start by understanding the dynamics of the problem. What does increasing n do? Let’s try with a few examples. If n 1, then yes, 61 − 1 5 is a multiple of 5. What does incrementing n to 2 look like? We get 62 − 1 35, which again is a multiple of 5. Next, n 3: but instead of just ﬁnding 63 − 1, what did the increase in n do? We will still subtract 1, but now we are multiplying by another 6 ﬁrst. Viewed another way, we are multiplying a number which is one more than a multiple of 5 by 6 (because 62 − 1 is a multiple of 5, so 62 is one more than a multiple of 5). What do numbers which are one more than a multiple of 5 look like? They must have last digit 1 or 6. What happens when you multiply such a number by 6? Depends on the number, but in any case, the last digit of the new number must be a 6. And then if you subtract 1, you get last digit 5, so a multiple of 5. The point is, every time we multiply by just one more six, we still get a number with last digit 6, so subtracting 1 gives us a multiple of 5. Now the formal proof: Proof. Let P(n) be the statement, “6n − 1 is a multiple of 5.” We will prove that P(n) is true for all n ∈ N. Base case: P(0) is true: 60 − 1 0 which is a multiple of 5. Inductive case: Let k be an arbitrary natural number. Assume, for induction, that P(k) is true. That is, 6k − 1 is a multiple of 5. Then 6k − 1 5j for some integer j. This means that 6k 5j + 1. Multiply both sides by 6: 6k+1 6(5j + 1) 30j + 6. 2.5. Induction 183 But we want to know about 6k+1 − 1, so subtract 1 from both sides: 6k+1 − 1 30j + 5. Of course 30j + 5 5(6j + 1), so is a multiple of 5. Therefore 6k+1 − 1 is a multiple of 5, or in other words, P(k + 1) is true. Thus, by the principle of mathematical induction P(n) is true for all n ∈ N. qed We had to be a little bit clever (i.e., use some algebra) to locate the − 1 inside of 6k+1 − 1 before we could apply the inductive hypothesis. This is what can make inductive proofs challenging. In the two examples above, we started with n 1 or n 0. We can start later if we need to. 6k Example 2.5.3 Prove that n 2 < 2n for all integers n ≥ 5. Solution. First, the idea of the argument. What happens when we increase n by 1? On the left-hand side, we increase the base of the square and go to the next square number. On the righthand side, we increase the power of 2. This means we double the number. So the question is, how does doubling a number relate to increasing to the next square? Think about what the diﬀerence of two consecutive squares looks like. We have (n + 1)2 − n 2 . This factors: (n + 1)2 − n 2 (n + 1 − n)(n + 1 + n) 2n + 1. But doubling the right-hand side increases it by 2n , since 2n+1 2n + 2n . When n is large enough, 2n > 2n + 1. What we are saying here is that each time n increases, the lefthand side grows by less than the right-hand side. So if the left-hand side starts smaller (as it does when n 5), it will never catch up. Now the formal proof: Proof. Let P(n) be the statement n 2 < 2n . We will prove P(n) is true for all integers n ≥ 5. Base case: P(5) is the statement 52 < 25 . Since 52 25 and 5 2 32, we see that P(5) is indeed true. Inductive case: Let k ≥ 5 be an arbitrary integer. Assume, for induction, that P(k) is true. That is, assume k 2 < 2k . We will prove that P(k + 1) is true, i.e., (k + 1)2 < 2k+1 . To prove such an inequality, 184 2. Sequences start with the left-hand side and work towards the right-hand side: (k + 1)2 k 2 + 2k + 1 < 2k + 2k + 1 < 2k + 2k . . . by the inductive hypothesis. . . . since 2k + 1 < 2k for k ≥ 5. 2k+1 . Following the equalities and inequalities through, we get (k + 1)2 < 2k+1 , in other words, P(k + 1). Therefore by the principle of mathematical induction, P(n) is true for all n ≥ 5. qed The previous example might remind you of the racetrack principle from calculus, which says that if f (a) < g(a), and f ′(x) < g ′(x) for x > a, then f (x) < g(x) for x > a. Same idea: the larger function is increasing at a faster rate than the smaller function, so the larger function will stay larger. In discrete math, we don’t have derivatives, so we look at diﬀerences. Thus induction is the way to go. Warning:. With great power, comes great responsibility. Induction isn’t magic. It seems very powerful to be able to assume P(k) is true. After all, we are trying to prove P(n) is true and the only diﬀerence is in the variable: k vs. n. Are we assuming that what we want to prove is true? Not really. We assume P(k) is true only for the sake of proving that P(k + 1) is true. Still you might start to believe that you can prove anything with induction. Consider this incorrect “proof” that every Canadian has the same eye color: Let P(n) be the statement that any n Canadians have the same eye color. P(1) is true, since everyone has the same eye color as themselves. Now assume P(k) is true. That is, assume that in any group of k Canadians, everyone has the same eye color. Now consider an arbitrary group of k + 1 Canadians. The ﬁrst k of these must all have the same eye color, since P(k) is true. Also, the last k of these must have the same eye color, since P(k) is true. So in fact, everyone the group must have the same eye color. Thus P(k + 1) is true. So by the principle of mathematical induction, P(n) is true for all n. Clearly something went wrong. The problem is that the proof that P(k) implies P(k + 1) assumes that k ≥ 2. We have only shown P(1) is true. In fact, P(2) is false. 2.5. Induction 185 Strong Induction Investigate! Start with a square piece of paper. You want to cut this square into smaller squares, leaving no waste (every piece of paper you end up with must be a square). Obviously it is possible to cut the square into 4 squares. You can also cut it into 9 squares. It turns out you can cut the square into 7 squares (although not all the same size). What other numbers of squares could you end up with? ! Attempt the above activity before proceeding ! Sometimes, to prove that P(k + 1) is true, it would be helpful to know that P(k) and P(k − 1) and P(k − 2) are all true. Consider the following puzzle: You have a rectangular chocolate bar, made up of n identical squares of chocolate. You can take such a bar and break it along any row or column. How many times will you have to break the bar to reduce it to n single chocolate squares? At ﬁrst, this question might seem impossible. Perhaps I meant to ask for the smallest number of breaks needed? Let’s investigate. Start with some small cases. If n 2, you must have a 1 × 2 rectangle, which can be reduced to single pieces in one break. With n 3, we must have a 1 × 3 bar, which requires two breaks: the ﬁrst break creates a single square and a 1 × 2 bar, which we know takes one (more) break. What about n 4? Now we could have a 2 × 2 bar, or a 1 × 4 bar. In the ﬁrst case, break the bar into two 2 × 2 bars, each which require one more break (that’s a total of three breaks required). If we started with a 1 × 4 bar, we have choices for our ﬁrst break. We could break the bar in half, creating two 1 × 2 bars, or we could break oﬀ a single square, leaving a 1 × 3 bar. But either way, we still need two more breaks, giving a total of three. It is starting to look like no matter how we break the bar (and no matter how the n squares are arranged into a rectangle), we will always have the same number of breaks required. It also looks like that number is one less than n: Conjecture 2.5.4 Given a n-square rectangular chocolate bar, it always takes n − 1 breaks to reduce the bar to single squares. It makes sense to prove this by induction because after breaking the bar once, you are left with smaller chocolate bars. Reducing to smaller cases is what induction is all about. We can inductively assume we already know 186 2. Sequences how to deal with these smaller bars. The problem is, if we are trying to prove the inductive case about a (k + 1)-square bar, we don’t know that after the ﬁrst break the remaining bar will have k squares. So we really need to assume that our conjecture is true for all cases less than k + 1. Is it valid to make this stronger assumption? Remember, in induction we are attempting to prove that P(n) is true for all n. What if that were not the case? Then there would be some ﬁrst n 0 for which P(n 0 ) was false. Since n 0 is the ﬁrst counterexample, we know that P(n) is true for all n < n 0 . Now we proceed to prove that P(n 0 ) is actually true, based on the assumption that P(n) is true for all smaller n. This is quite an advantage: we now have a stronger inductive hypothesis. We can assume that P(1), P(2), P(3), . . . P(k) is true, just to show that P(k + 1) is true. Previously, we just assumed P(k) for this purpose. It is slightly easier if we change our variables for strong induction. Here is what the formal proof would look like: Strong Induction Proof Structure. Again, start by saying what you want to prove: “Let P(n) be the statement. . . ” Then establish two facts: 1. Base case: Prove that P(0) is true. 2. Inductive case: Assume P(k) is true for all k < n. Prove that P(n) is true. Conclude, “therefore, by strong induction, P(n) is true for all n > 0.” Of course, it is acceptable to replace 0 with a larger base case if needed.5 Let’s prove our conjecture about the chocolate bar puzzle: Proof. Let P(n) be the statement, “it takes n −1 breaks to reduce a n-square chocolate bar to single squares.” Base case: Consider P(2). The squares must be arranged into a 1 × 2 rectangle, and we require 2 − 1 1 breaks to reduce this to single squares. Inductive case: Fix an arbitrary n ≥ 2 and assume P(k) is true for all k < n. Consider a n-square rectangular chocolate bar. Break the bar once along any row or column. This results in two chocolate bars, say of sizes a and b. That is, we have an a-square rectangular chocolate bar, a b-square rectangular chocolate bar, and a + b n. We also know that a < n and b < n, so by our inductive hypothesis, P(a) and P(b) are true. To reduce the a-sqaure bar to single squares takes a −1 breaks; to reduce the b-square bar to single squares takes b −1 breaks. 5Technically, strong induction does not require you to prove a separate base case. This is because when proving the inductive case, you must show that P(0) is true, assuming P(k) is true for all k < 0. But this is not any help so you end up proving P(0) anyway. To be on the safe side, we will always include the base case separately. 2.5. Induction 187 Doing this results in our original bar being reduced to single squares. All together it took the initial break, plus the a − 1 and b − 1 breaks, for a total of 1 + a − 1 + b − 1 a + b − 1 n − 1 breaks. Thus P(n) is true. Therefore, by strong induction, P(n) is true for all n ≥ 2. qed Here is a more mathematically relevant example: Example 2.5.5 Prove that any natural number greater than 1 is either prime or can be written as the product of primes. Solution. First, the idea: if we take some number n, maybe it is prime. If so, we are done. If not, then it is composite, so it is the product of two smaller numbers. Each of these factors is smaller than n (but at least 2), so we can repeat the argument with these numbers. We have reduced to a smaller case. Now the formal proof: Proof. Let P(n) be the statement, “n is either prime or can be written as the product of primes.” We will prove P(n) is true for all n ≥ 2. Base case: P(2) is true because 2 is indeed prime. Inductive case: assume P(k) is true for all k < n. We want to show that P(n) is true. That is, we want to show that n is either prime or is the product of primes. If n is prime, we are done. If not, then n has more than 2 divisors, so we can write n m 1 · m2 , with m1 and m 2 less than n (and greater than 1). By the inductive hypothesis, m1 and m 2 are each either prime or can be written as the product of primes. In either case, we have that n is written as the product of primes. Thus by the strong induction, P(n) is true for all n ≥ 2. qed Whether you use regular induction or strong induction depends on the statement you want to prove. If you wanted to be safe, you could always use strong induction. It really is stronger, so can accomplish everything “weak” induction can. That said, using regular induction is often easier since there is only one place you can use the induction hypothesis. There is also something to be said for elegance in proofs. If you can prove a statement using simpler tools, it is nice to do so. As a ﬁnal contrast between the two forms of induction, consider once more the stamp problem. Regular induction worked by showing how to increase postage by one cent (either replacing three 5-cent stamps with two 8-cent stamps, or three 8-cent stamps with ﬁve 5-cent stamps). We could give a slightly diﬀerent proof using strong induction. First, we could show ﬁve base cases: it is possible to make 28, 29, 30, 31, and 32 cents (we would actually say how each of these is made). Now assume 188 2. Sequences that it is possible to make k cents of postage for all k < n as long as k ≥ 28. As long as n > 32, this means in particular we can make k n − 5 cents. Now add a 5-cent stamp to get make n cents. Exercises 1. On the way to the market, you exchange your cow for some magic dark chocolate espresso beans. These beans have the property that every night at midnight, each bean splits into two, eﬀectively doubling your collection. You decide to take advantage of this and each morning (around 8am) you eat 5 beans. (a) Explain why it is true that if at noon on day n you have a number of beans ending in a 5, then at noon on day n + 1 you will still have a number of beans ending in a 5. (b) Why is the previous fact not enough to conclude that you will always have a number of beans ending in a 5? What additional fact would you need? (c) Assuming you have the additional fact in part (b), and have successfully proved the fact in part (a), how do you know that you will always have a number of beans ending in a 5? Illustrate what is going on by carefully explaining how the two facts above prove that you will have a number of beans ending in a 5 on day 4 speciﬁcally. In other words, explain why induction works in this context. 2. Use induction to prove for all n ∈ N that n Õ 2k 2n+1 − 1. k0 7n − 1 is a multiple of 6 for all n ∈ N. 3. Prove that 4. Prove that 1 + 3 + 5 + · · · + (2n − 1) n 2 for all n ≥ 1. 5. Prove that F0 + F2 + F4 + · · · + F2n F2n+1 − 1 where Fn is the nth Fibonacci number. 6. Prove that 2n < n! for all n ≥ 4. (Recall, n! 1 · 2 · 3 · · · · · n.) 7. Prove, by mathematical induction, that F0 +F1 +F2 +· · ·+Fn Fn+2 −1, where Fn is the nth Fibonacci number (F0 0, F1 1 and Fn Fn−1 + Fn−2 ). 8. Zombie Euler and Zombie Cauchy, two famous zombie mathematicians, have just signed up for Twitter accounts. After one day, Zombie Cauchy has more followers than Zombie Euler. Each day after that, the number of new followers of Zombie Cauchy is exactly the same as 2.5. Induction 189 the number of new followers of Zombie Euler (and neither lose any followers). Explain how a proof by mathematical induction can show that on every day after the ﬁrst day, Zombie Cauchy will have more followers than Zombie Euler. That is, explain what the base case and inductive case are, and why they together prove that Zombie Cauchy will have more followers on the 4th day. 9. Find the largest number of points which a football team cannot get exactly using just 3-point ﬁeld goals and 7-point touchdowns (ignore the possibilities of safeties, missed extra points, and two point conversions). Prove your answer is correct by mathematical induction. 10. Prove that the sum of n squares can be found as follows 12 + 22 + 32 + ... + n 2 n(n + 1)(2n + 1) . 6 11. Prove that the sum of the interior angles of a convex n-gon is (n − 2) · 180◦ . (A convex n-gon is a polygon with n sides for which each interior angle is less than 180◦ .) 12. What is wrong with the following “proof” of the “fact” that n+3 n+7 for all values of n (besides of course that the thing it is claiming to prove is false)? Proof. Let P(n) be the statement that n + 3 n + 7. We will prove that P(n) is true for all n ∈ N. Assume, for induction that P(k) is true. That is, k + 3 k + 7. We must show that P(k + 1) is true. Now since k + 3 k + 7, add 1 to both sides. This gives k + 3 + 1 k + 7 + 1. Regrouping (k + 1) + 3 (k + 1) + 7. But this is simply P(k + 1). Thus by the principle of mathematical induction P(n) is true for all n ∈ N. qed 13. The proof in the previous problem does not work. But if we modify the “fact,” we can get a working proof. Prove that n + 3 < n + 7 for all values of n ∈ N. You can do this proof with algebra (without induction), but the goal of this exercise is to write out a valid induction proof. 14. Find the ﬂaw in the following “proof” of the “fact” that n < 100 for every n ∈ N. Proof. Let P(n) be the statement n < 100. We will prove P(n) is true for all n ∈ N. First we establish the base case: when n 0, P(n) is true, because 0 < 100. Now for the inductive step, assume P(k) is true. That is, k < 100. Now if k < 100, then k is some number, like 80. Of course 80+1 81 which is still less than 100. So k +1 < 100 as well. But this is what P(k +1) claims, so we have shown that P(k) → P(k +1). 190 2. Sequences Thus by the principle of mathematical induction, P(n) is true for all n ∈ N. qed 15. While the above proof does not work (it better not since the statement it is trying to prove is false!) we can prove something similar. Prove that there is a strictly increasing sequence a 1 , a 2 , a 3 , . . . of numbers (not necessarily integers) such that a n < 100 for all n ∈ N. (By strictly increasing we mean a n < a n+1 for all n. So each term must be larger than the last.) 16. What is wrong with the following “proof” of the “fact” that for all n ∈ N, the number n 2 + n is odd? Proof. Let P(n) be the statement “n 2 + n is odd.” We will prove that P(n) is true for all n ∈ N. Suppose for induction that P(k) is true, that is, that k 2 + k is odd. Now consider the statement P(k + 1). Now (k + 1)2 + (k + 1) k 2 + 2k + 1 + k + 1 k 2 + k + 2k + 2. By the inductive hypothesis, k 2 + k is odd, and of course 2k + 2 is even. An odd plus an even is always odd, so therefore (k + 1)2 + (k + 1) is odd. Therefore by the principle of mathematical induction, P(n) is true for all n ∈ N. qed 17. Now give a valid proof (by induction, even though you might be able to do so without using induction) of the statement, “for all n ∈ N, the number n 2 + n is even.” 18. Prove that there is a sequence of positive real numbers a 0 , a 1 , a2 , . . . such that the partial sum a 0 + a 1 + a2 + · · · + a n is strictly less than 2 for all n ∈ N. Hint: think about how you could deﬁne what a k+1 is to make the induction argument work. 19. Prove that every positive integer is either a power of 2, or can be written as the sum of distinct powers of 2. 20. Prove, using strong induction, that every natural number is either a Fibonacci number or can be written as the sum of distinct Fibonacci numbers. 21. Use induction to prove that if n people all shake hands with each n(n−1) other, that the total number of handshakes is 2 . 22. Suppose that a particular real number x has the property that x + x1 is an integer. Prove that x n + x1n is an integer for all natural numbers n. 23. Use induction to prove that n Õ n k0 row of Pascal’s Triangle is 2n . k 2n . That is, the sum of the nth 2.5. Induction 24. Use induction to prove 40 + 51 + 62 + · · · + example of the hockey stick theorem.) 4+n n 5+n n . 191 (This is an 25. Use the product rule for logarithms (log(ab) log(a)+log(b)) to prove, by induction on n, that log(a n ) n log(a), for all natural numbers n ≥ 2. 26. Let f1 , f2 , . . . , f n be diﬀerentiable functions. Prove, using induction, that ( f1 + f2 + · · · + f n )′ f1′ + f2′ + · · · + f n′ . You may assume ( f + g)′ f ′ + g ′ for any diﬀerentiable functions f and g. 27. Suppose f1 , f2 , . . . , f n are diﬀerentiable functions. Use mathematical induction to prove the generalized product rule: ( f1 f2 f3 · · · f n )′ f1′ f2 f3 · · · f n + f1 f2′ f3 · · · f n + f1 f2 f3′ · · · f n +· · ·+ f1 f2 f3 · · · f n′ . You may assume the product rule for two functions is true. 28. You will prove that the Fibonacci numbers satisfy the identity Fn2 + 2 Fn+1 F2n+1 . One way to do this is to prove the more general identity, Fm Fn + Fm+1 Fn+1 Fm+n+1 , and realize that when m n we get our desired result. Note that we now have two variables, so we want to prove this for all m ≥ 0 and all n ≥ 0 at the same time. For each such pair (m, n), let P(m, n) be the statement Fm Fn + Fm+1 Fn+1 Fm+n+1 (a) First ﬁx m 0 and give a proof by mathematical induction that P(0, n) holds for all n ≥ 0. Note this proof will be very easy. (b) Now ﬁx an arbitrary n and give a proof by strong mathematical induction that P(m, n) holds for all m ≥ 0. (c) You can now conclude that P(m, n) holds for all m, n ≥ 0. Do you believe that? Explain why this sort of induction is valid. For example, why do your proofs above guarantee that P(2, 3) is true? 29. Given a square, you can cut the square into smaller squares by cutting along lines parallel to the sides of the original square (these lines do not need to travel the entire side length of the original square). For example, by cutting along the lines below, you will divide a square into 6 smaller squares: 192 2. Sequences Prove, using strong induction, that it is possible to cut a square into n smaller squares for any n ≥ 6. 2.6. Chapter Summary 2.6 193 Chapter Summary Investigate! Each day your supply of magic chocolate covered espresso beans doubles (each one splits in half), but then you eat 5 of them. You have 10 at the start of day 0. 1. Write out the ﬁrst few terms of the sequence. Then give a recursive deﬁnition for the sequence and explain how you know it is correct. 2. Prove, using induction, that the last digit of the number of beans you have on the nth day is always a 5 for all n ≥ 1. 3. Find a closed formula for the nth term of the sequence and prove it is correct by induction. ! Attempt the above activity before proceeding ! In this chapter we explored sequences and mathematical induction. At ﬁrst these might not seem entirely related, but there is a link: recursive reasoning. When we have many cases (maybe inﬁnitely many), it is often easier to describe a particular case by saying how it relates to other cases, instead of describing it absolutely. For sequences, we can describe the nth term in the sequence by saying how it is related to the previous term. When showing a statement involving the variable n is true for all values of n, we can describe why the case for n k is true on the basis of why the case for n k − 1 is true. While thinking of problems recursively is often easer than thinking of them absolutely (at least after you get used to thinking in this way), our ultimate goal is to move beyond this recursive description. For sequences, we want to ﬁnd closed formulas for the nth term of the sequence. For proofs, we want to know the statement is actually true for a particular n (not only under the assumption that the statement is true for the previous value of n). In this chapter we saw some methods for moving from recursive descriptions to absolute descriptions. • If the terms of a sequence increase by a constant diﬀerence or constant ratio (these are both recursive descriptions), then the sequences are arithmetic or geometric, respectively, and we have closed formulas for each of these based on the initial terms and common diﬀerence or ratio. • If the terms of a sequence increase at a polynomial rate (that is, if the diﬀerences between terms form a sequence with a polynomial closed 194 2. Sequences formula), then the sequence is itself given by a polynomial closed formula (of degree one more than the sequence of diﬀerences). • If the terms of a sequence increase at an exponential rate, then we expect the closed formula for the sequence to be exponential. These sequences often have relatively nice recursive formulas, and the characteristic root technique allows us to ﬁnd the closed formula for these sequences. • If we want to prove that a statement is true for all values of n (greater than some ﬁrst small value), and we can describe why the statement being true for n k implies the statement is true for n k + 1, then the principle of mathematical induction gives us that the statement is true for all values of n (greater than the base case). Throughout the chapter we tried to understand why these facts listed above are true. In part, that is what proofs, by induction or not, attempt to accomplish: they explain why mathematical truths are in fact truths. As we develop our ability to reason about mathematics, it is a good idea to make sure that the methods of our reasoning are sound. The branch of mathematics that deals with deciding whether reasoning is good or not is mathematical logic, the subject of the next chapter. Chapter Review 1. Find 3 + 7 + 11 + · · · + 427. 2. Consider the sequence 2, 6, 10, 14, . . . , 4n + 6. (a) How many terms are there in the sequence? (b) What is the second-to-last term? (c) Find the sum of all the terms in the sequence. 3. Consider the sequence given by a n 2 · 5n−1 . (a) Find the ﬁrst 4 terms of the sequence. What sort of sequence is this? (b) Find the sum of the ﬁrst 25 terms. That is, compute 4. Í25 k1 ak . Consider the sequence 5, 11, 19, 29, 41, 55, . . .. Assume a 1 5. (a) Find a closed formula for a n , the nth term of the sequence, by writing each term as a sum of a sequence. Hint: ﬁrst ﬁnd a 0 , but ignore it when collapsing the sum. (b) Find a closed formula again, this time using either polynomial ﬁtting or the characteristic root technique (whichever is appropriate). Show your work. 2.6. Chapter Summary 195 (c) Find a closed formula once again, this time by recognizing the sequence as a modiﬁcation to some well known sequence(s). Explain. 5. Use polynomial ﬁtting to ﬁnd a closed formula for the sequence (a n )n≥1 : 4, 11, 20, 31, 44, . . . . 6. Suppose the closed formula for a particular sequence is a degree 3 polynomial. What can you say about the closed formula for: (a) The sequence of partial sums. (b) The sequence of second diﬀerences. 7. Consider the sequence given recursively by a 1 4, a 2 6 and a n a n−1 + a n−2 . (a) Write out the ﬁrst 6 terms of the sequence. (b) Could the closed formula for a n be a polynomial? Explain. 8. The sequence (a n )n≥1 starts −1, 0, 2, 5, 9, 14 . . . and has closed formula (n + 1)(n − 2) . Use this fact to ﬁnd a closed formula for the an 2 sequence (b n )n≥1 which starts 4, 10, 18, 28, 40, . . . . 9. The in song The Twelve Days of Christmas, my true love gave to me ﬁrst 1 gift, then 2 gifts and 1 gift, then 3 gifts, 2 gifts and 1 gift, and so on. How many gifts did my true love give me all together during the twelve days? 10. Consider the recurrence relation a n 3a n−1 + 10a n−2 with ﬁrst two terms a 0 1 and a 1 2. (a) Write out the ﬁrst 5 terms of the sequence deﬁned by this recurrence relation. (b) Solve the recurrence relation. That is, ﬁnd a closed formula for an . 11. Consider the recurrence relation a n 2a n−1 + 8a n−2 , with initial terms a 0 1 and a1 3. (a) Find the next two terms of the sequence (a 2 and a 3 ). (b) Solve the recurrence relation. That is, ﬁnd a closed formula for the nth term of the sequence. 12. Your magic chocolate bunnies reproduce like rabbits: every large bunny produces 2 new mini bunnies each day, and each day every 196 2. Sequences mini bunny born the previous day grows into a large bunny. Assume you start with 2 mini bunnies and no bunny ever dies (or gets eaten). (a) Write out the ﬁrst few terms of the sequence. (b) Give a recursive deﬁnition of the sequence and explain why it is correct. (c) Find a closed formula for the nth term of the sequence. 13. Consider the sequence of partial sums of squares of Fibonacci numbers: F12 , F12 + F22 , F12 + F22 + F32 , . . .. The sequences starts 1, 2, 6, 15, 40, . . . (a) Guess a formula for the nth partial sum, in terms of Fibonacci numbers. Hint: write each term as a product. (b) Prove your formula is correct by mathematical induction. (c) Explain what this problem has to do with the following picture: 14. Prove the following statements by mathematical induction: (a) n! < n n for n ≥ 2 (b) 1 1 1 1 n + + +···+ for all n ∈ Z+ . 1·2 2·3 3·4 n · (n + 1) n + 1 (c) 4n − 1 is a multiple of 3 for all n ∈ N. (d) The greatest amount of postage you cannot make exactly using 4 and 9 cent stamps is 23 cents. (e) Every even number squared is divisible by 4. 15. Prove 13 + 23 + 33 + ··· + n3 n(n+1) 2 2 holds for all n ≥ 1, by mathematical induction. 16. Suppose a 0 1, a 1 1 and a n 3a n−1 − 2a n−1 . Prove, using strong induction, that a n 1 for all n. 17. Prove using induction that every set containing n elements has 2n diﬀerent subsets for any n ≥ 1. Chapter 3 Symbolic Logic and Proofs Logic is the study of consequence. Given a few mathematical statements or facts, we would like to be able to draw some conclusions. For example, if I told you that a particular real-valued function was continuous on the interval [0, 1], and f (0) −1 and f (1) 5, can we conclude that there is some point between [0, 1] where the graph of the function crosses the x-axis? Yes, we can, thanks to the Intermediate Value Theorem from Calculus. Can we conclude that there is exactly one point? No. Whenever we ﬁnd an “answer” in math, we really have a (perhaps hidden) argument. Mathematics is really about proving general statements (like the Intermediate Value Theorem), and this too is done via an argument, usually called a proof. We start with some given conditions, the premises of our argument, and from these we ﬁnd a consequence of interest, our conclusion. The problem is, as you no doubt know from arguing with friends, not all arguments are good arguments. A “bad” argument is one in which the conclusion does not follow from the premises, i.e., the conclusion is not a consequence of the premises. Logic is the study of what makes an argument good or bad. In other words, logic aims to determine in which cases a conclusion is, or is not, a consequence of a set of premises. By the way, “argument” is actually a technical term in math (and philosophy, another discipline which studies logic): Arguments. An argument is a set of statements, one of which is called the conclusion and the rest of which are called premises. An argument is said to be valid if the conclusion must be true whenever the premises are all true. An argument is invalid if it is not valid; it is possible for all the premises to be true and the conclusion to be false. For example, consider the following two arguments: ∴ ∴ If Edith eats her vegetables, then she can have a cookie. Edith eats her vegetables. Edith gets a cookie. Florence must eat her vegetables in order to get a cookie. Florence eats her vegetables. Florence gets a cookie. 197 198 3. Symbolic Logic and Proofs (The symbol “ ∴ ” means “therefore”) Are these arguments valid? Hopefully you agree that the ﬁrst one is but the second one is not. Logic tells us why by analyzing the structure of the statements in the argument. Notice the two arguments above look almost identical. Edith and Florence both eat their vegetables. In both cases there is a connection between the eating of vegetables and cookies. But we claim that it is valid to conclude that Edith gets a cookie, but not that Florence does. The diﬀerence must be in the connection between eating vegetables and getting cookies. We need to be skilled at reading and comprehending these sentences. Do the two sentences mean the same thing? Unfortunately, in everyday language we are often sloppy, and you might be tempted to say they are equivalent. But notice that just because Florence must eat her vegetables, we have not said that doing so would be enough (she might also need to clean her room, for example). In everyday (non-mathematical) practice, you might be tempted to say this “other direction” is implied. In mathematics, we never get that luxury. Before proceeding, it might be a good idea to quickly review Section 0.2 where we ﬁrst encountered statements and the various forms they can take. The goal now is to see what mathematical tools we can develop to better analyze these, and then to see how this helps read and write proofs. 3.1 Propositional Logic Investigate! You stumble upon two trolls playing Stratego®. They tell you: Troll 1: If we are cousins, then we are both knaves. Troll 2: We are cousins or we are both knaves. Could both trolls be knights? Recall that all trolls are either always-truth-telling knights or always-lying knaves. ! Attempt the above activity before proceeding ! A proposition is simply a statement. Propositional logic studies the ways statements can interact with each other. It is important to remember that propositional logic does not really care about the content of the statements. For example, in terms of propositional logic, the claims, “if the moon is made of cheese then basketballs are round,” and “if spiders have eight legs then Sam walks with a limp” are exactly the same. They are both implications: statements of the form, P → Q. 3.1. Propositional Logic 199 Truth Tables Here’s a question about playing Monopoly: If you get more doubles than any other player then you will lose, or if you lose then you must have bought the most properties. True or false? We will answer this question, and won’t need to know anything about Monopoly. Instead we will look at the logical form of the statement. We need to decide when the statement (P → Q) ∨ (Q → R) is true. Using the deﬁnitions of the connectives in Section 0.2, we see that for this to be true, either P → Q must be true or Q → R must be true (or both). Those are true if either P is false or Q is true (in the ﬁrst case) and Q is false or R is true (in the second case). So—yeah, it gets kind of messy. Luckily, we can make a chart to keep track of all the possibilities. Enter truth tables. The idea is this: on each row, we list a possible combination of T’s and F’s (for true and false) for each of the sentential variables, and then mark down whether the statement in question is true or false in that case. We do this for every possible combination of T’s and F’s. Then we can clearly see in which cases the statement is true or false. For complicated statements, we will ﬁrst ﬁll in values for each part of the statement, as a way of breaking up our task into smaller, more manageable pieces. Since the truth value of a statement is completely determined by the truth values of its parts and how they are connected, all you really need to know is the truth tables for each of the logical connectives. Here they are: P T T F F Q T F T F P∧Q T F F F P T T F F Q T F T F P∨Q T T T F P T T F F Q T F T F P→Q T F T T P T T F F Q T F T F P↔Q T F F T The truth table for negation looks like this: P T F ¬P F T None of these truth tables should come as a surprise; they are all just restating the deﬁnitions of the connectives. Let’s try another one. 200 3. Symbolic Logic and Proofs Example 3.1.1 Make a truth table for the statement ¬P ∨ Q. Solution. Note that this statement is not ¬(P ∨ Q), the negation belongs to P alone. Here is the truth table: P T T F F Q T F T F ¬P F F T T ¬P ∨ Q T F T T We added a column for ¬P to make ﬁlling out the last column easier. The entries in the ¬P column were determined by the entries in the P column. Then to ﬁll in the ﬁnal column, look only at the column for Q and the column for ¬P and use the rule for ∨. Now let’s answer our question about monopoly: Example 3.1.2 Analyze the statement, “if you get more doubles than any other player you will lose, or that if you lose you must have bought the most properties,” using truth tables. Solution. Represent the statement in symbols as (P → Q) ∨ (Q → R), where P is the statement “you get more doubles than any other player,” Q is the statement “you will lose,” and R is the statement “you must have bought the most properties.” Now make a truth table. The truth table needs to contain 8 rows in order to account for every possible combination of truth and falsity among the three statements. Here is the full truth table: P T T T T F F F F Q T T F F T T F F R T F T F T F T F P→Q T T F F T T T T Q→R T F T T T F T T (P → Q) ∨ (Q → R) T T T T T T T T 3.1. Propositional Logic 201 The ﬁrst three columns are simply a systematic listing of all possible combinations of T and F for the three statements (do you see how you would list the 16 possible combinations for four statements?). The next two columns are determined by the values of P, Q, and R and the deﬁnition of implication. Then, the last column is determined by the values in the previous two columns and the deﬁnition of ∨. It is this ﬁnal column we care about. Notice that in each of the eight possible cases, the statement in question is true. So our statement about monopoly is true (regardless of how many properties you own, how many doubles you roll, or whether you win or lose). The statement about monopoly is an example of a tautology, a statement which is true on the basis of its logical form alone. Tautologies are always true but they don’t tell us much about the world. No knowledge about monopoly was required to determine that the statement was true. In fact, it is equally true that “If the moon is made of cheese, then Elvis is still alive, or if Elvis is still alive, then unicorns have 5 legs.” Logical Equivalence You might have noticed in Example 3.1.1 that the ﬁnal column in the truth table for ¬P ∨ Q is identical to the ﬁnal column in the truth table for P → Q: P T T F F Q T F T F P→Q T F T T ¬P ∨ Q T F T T This says that no matter what P and Q are, the statements ¬P ∨ Q and P → Q either both true or both false. We therefore say these statements are logically equivalent. Logical Equivalence. Two (molecular) statements P and Q are logically equivalent provided P is true precisely when Q is true. That is, P and Q have the same truth value under any assignment of truth values to their atomic parts. To verify that two statements are logically equivalent, you can make a truth table for each and check whether the columns for the two statements are identical. 202 3. Symbolic Logic and Proofs Recognizing two statements as logically equivalent can be very helpful. Rephrasing a mathematical statement can often lend insight into what it is saying, or how to prove or refute it. By using truth tables we can systematically verify that two statements are indeed logically equivalent. Example 3.1.3 Are the statements, “it will not rain or snow” and “it will not rain and it will not snow” logically equivalent? Solution. We want to know whether ¬(P∨Q) is logically equivalent to ¬P ∧ ¬Q. Make a truth table which includes both statements: P T T F F Q T F T F ¬(P ∨ Q) F F F T ¬P ∧ ¬Q F F F T Since in every row the truth values for the two statements are equal, the two statements are logically equivalent. Notice that this example gives us a way to “distribute” a negation over a disjunction (an “or”). We have a similar rule for distributing over conjunctions (“and”s): De Morgan’s Laws. ¬(P ∧ Q) is logically equivalent to ¬P ∨ ¬Q. ¬(P ∨ Q) is logically equivalent to ¬P ∧ ¬Q. This suggests there might be a sort of “algebra” you could apply to statements (okay, there is: it is called Boolean algebra) to transform one statement into another. We can start collecting useful examples of logical equivalence, and apply them in succession to a statement, instead of writing out a complicated truth table. De Morgan’s laws do not do not directly help us with implications, but as we saw above, every implication can be written as a disjunction: Implications are Disjuctions. P → Q is logically equivalent to ¬P ∨ Q. Example: “If a number is a multiple of 4, then it is even” is equivalent to, “a number is not a multiple of 4 or (else) it is even.” 3.1. Propositional Logic 203 With this and De Morgan’s laws, you can take any statement and simplify it to the point where negations are only being applied to atomic propositions. Well, actually not, because you could get multiple negations stacked up. But this can be easily dealt with: Double Negation. ¬¬P is logically equivalent to P. Example: “It is not the case that c is not odd” means “c is odd.” Let’s see how we can apply the equivalences we have encountered so far. Example 3.1.4 Prove that the statements ¬(P → Q) and P ∧ ¬Q are logically equivalent without using truth tables. Solution. We want to start with one of the statements, and transform it into the other through a sequence of logically equivalent statements. Start with ¬(P → Q). We can rewrite the implication as a disjunction this is logically equivalent to ¬(¬P ∨ Q). Now apply DeMorgan’s law to get ¬¬P ∧ ¬Q. Finally, use double negation to arrive at P ∧ ¬Q Notice that the above example illustrates that the negation of an implication is NOT an implication: it is a conjunction! We saw this before, in Section 0.2, but it is so important and useful, it warants a second blue box here: Negation of an Implication. The negation of an implication is a conjuction: ¬(P → Q) is logically equivalent to P ∧ ¬Q. That is, the only way for an implication to be false is for the hypothesis to be true AND the conclusion to be false. To verify that two statements are logically equivalent, you can use truth tables or a sequence of logically equivalent replacements. The truth 204 3. Symbolic Logic and Proofs table method, although cumbersome, has the advantage that it can verify that two statements are NOT logically equivalent. Example 3.1.5 Are the statements (P ∨ Q) → R and (P → R) ∨ (Q → R) logically equivalent? Solution. Note that while we could start rewriting these statements with logically equivalent replacements in the hopes of transforming one into another, we will never be sure that our failure is due to their lack of logical equivalence rather than our lack of imagination. So instead, let’s make a truth table: P T T T T F F F F Q T T F F T T F F R T F T F T F T F (P ∨ Q) → R T F T F T F T T (P → R) ∨ (Q → R) T F T T T T T T Look at the fourth (or sixth) row. In this case, (P → R)∨(Q → R) is true, but (P ∨ Q) → R is false. Therefore the statements are not logically equivalent. While we don’t have logical equivalence, it is the case that whenever (P ∨ Q) → R is true, so is (P → R) ∨ (Q → R). This tells us that we can deduce (P → R) ∨ (Q → R) from (P ∨ Q) → R, just not the reverse direction. Deductions Investigate! Holmes owns two suits: one black and one tweed. He always wears either a tweed suit or sandals. Whenever he wears his tweed suit and a purple shirt, he chooses to not wear a tie. He never wears the tweed suit unless he is also wearing either a purple shirt or sandals. Whenever he wears sandals, he also wears a purple shirt. Yesterday, Holmes wore a bow tie. What else did he wear? ! Attempt the above activity before proceeding ! 3.1. Propositional Logic 205 Earlier we claimed that the following was a valid argument: If Edith eats her vegetables, then she can have a cookie. Edith ate her vegetables. Therefore Edith gets a cookie. How do we know this is valid? Let’s look at the form of the statements. Let P denote “Edith eats her vegetables” and Q denote “Edith can have a cookie.” The logical form of the argument is then: P→Q P Q ∴ This is an example of a deduction rule, an argument form which is always valid. This one is a particularly famous rule called modus ponens. Are you convinced that it is a valid deduction rule? If not, consider the following truth table: P T T F F Q T F T F P→Q T F T T This is just the truth table for P → Q, but what matters here is that all the lines in the deduction rule have their own column in the truth table. Remember that an argument is valid provided the conclusion must be true given that the premises are true. The premises in this case are P → Q and P. Which rows of the truth table correspond to both of these being true? P is true in the ﬁrst two rows, and of those, only the ﬁrst row has P → Q true as well. And lo-and-behold, in this one case, Q is also true. So if P → Q and P are both true, we see that Q must be true as well. Here are a few more examples. Example 3.1.6 Show that ∴ P→Q ¬P → Q Q is a valid deduction rule. Solution. We make a truth table which contains all the lines of the argument form: 206 3. Symbolic Logic and Proofs P T T F F P→Q T F T T Q T F T F ¬P F F T T ¬P → Q T T T F (we include a column for ¬P just as a step to help getting the column for ¬P → Q). Now look at all the rows for which both P → Q and ¬P → Q are true. This happens only in rows 1 and 3. Hey! In those rows Q is true as well, so the argument form is valid (it is a valid deduction rule). Example 3.1.7 Decide whether ∴ P→R Q→R R P∨Q is a valid deduction rule. Solution. Let’s make a truth table containing all four statements. P T T T T F F F F Q T T F F T T F F R T F T F T F T F P→R T F T F T T T T Q→R T F T T T F T T P∨Q T T T T T T F F Look at the second to last row. Here all three premises of the argument are true, but the conclusion is false. Thus this is not a valid deduction rule. While we have the truth table in front of us, look at rows 1, 3, and 5. These are the only rows in which all of the statements statements P → R, Q → R, and P ∨ Q are true. It also happens that R is true in these rows as well. Thus we have discovered a new deduction rule we know is valid: 3.1. Propositional Logic ∴ 207 P→R Q→R P∨Q R Beyond Propositions As we saw in Section 0.2, not every statement can be analyzed using logical connectives alone. For example, we might want to work with the statement: All primes greater than 2 are odd. To write this statement symbolically, we must use quantiﬁers. We can translate as follows: ∀x((P(x) ∧ x > 2) → O(x)). In this case, we are using P(x) to denote “x is prime” and O(x) to denote “x is odd.” These are not propositions, since their truth value depends on the input x. Better to think of P and O as denoting properties of their input. The technical term for these is predicates and when we study them in logic, we need to use predicate logic. It is important to stress that predicate logic extends propositional logic (much in the way quantum mechanics extends classical mechanics). You will notice that our statement above still used the (propositional) logical connectives. Everything that we learned about logical equivalence and deductions still applies. However, predicate logic allows us to analyze statements at a higher resolution, digging down into the individual propositions P, Q, etc. A full treatment of predicate logic is beyond the scope of this text. One reason is that there is no systematic procedure for deciding whether two statements in predicate logic are logically equivalent (i.e., there is no analogue to truth tables here). Rather, we end with a two examples of logical equivalence and deduction, to pique your interest. Example 3.1.8 Suppose we claim that there is no smallest number. We can translate this into symbols as ¬∃x∀y(x ≤ y) 208 3. Symbolic Logic and Proofs (literally, “it is not true that there is a number x such that for all numbers y, x is less than or equal to y”). However, we know how negation interacts with quantiﬁers: we can pass a negation over a quantiﬁer by switching the quantiﬁer type (between universal and existential). So the statement above should be logically equivalent to ∀x∃y(y < x). Notice that y < x is the negation of x ≤ y. This literally says, “for every number x there is a number y which is smaller than x.” We see that this is another way to make our original claim. Example 3.1.9 Can you switch the order of quantiﬁers? For example, consider the two statements: ∀x∃yP(x, y) and ∃y∀xP(x, y). Are these logically equivalent? Solution. These statements are NOT logically equivalent. To see this, we should provide an interpretation of the predicate P(x, y) which makes one of the statements true and the other false. Let P(x, y) be the predicate x < y. It is true, in the natural numbers, that for all x there is some y greater than that x (since there are inﬁnitely many numbers). However, there is not a natural number y which is greater than every number x. Thus it is possible for ∀x∃yP(x, y) to be true while ∃y∀xP(x, y) is false. We cannot do the reverse of this though. If there is some y for which every x satisﬁes P(x, y), then certainly for every x there is some y which satisﬁes P(x, y). The ﬁrst is saying we can ﬁnd one y that works for every x. The second allows diﬀerent y’s to work for diﬀerent x’s, but there is nothing preventing us from using the same y that work for every x. In other words, while we don’t have logical equivalence between the two statements, we do have a valid deduction rule: ∴ ∃y∀xP(x, y) ∀x∃yP(x, y) Put yet another way, this says that the single statement ∃y∀xP(x, y) → ∀x∃yP(x, y) 3.1. Propositional Logic 209 is always true. This is sort of like a tautology, although we reserve that term for necessary truths in propositional logic. A statement in predicate logic that is necessarily true gets the more prestigious designation of a law of logic (or sometimes logically valid, but that is less fun). Exercises 1. Consider the statement about a party, “If it’s your birthday or there will be cake, then there will be cake.” (a) Translate the above statement into symbols. Clearly state which statement is P and which is Q. (b) Make a truth table for the statement. (c) Assuming the statement is true, what (if anything) can you conclude if there will be cake? (d) Assuming the statement is true, what (if anything) can you conclude if there will not be cake? (e) Suppose you found out that the statement was a lie. What can you conclude? 2. Make a truth table for the statement (P ∨ Q) → (P ∧ Q). 3. Make a truth table for the statement ¬P ∧ (Q → P). What can you conclude about P and Q if you know the statement is true? 4. Make a truth table for the statement ¬P → (Q ∧ R). 5. Geoﬀ Poshingten is out at a fancy pizza joint, and decides to order a calzone. When the waiter asks what he would like in it, he replies, “I want either pepperoni or sausage. Also, if I have sausage, then I must also include quail. Oh, and if I have pepperoni or quail then I must also have ricotta cheese.” (a) Translate Geoﬀ’s order into logical symbols. (b) The waiter knows that Geoﬀ is either a liar or a truth-teller (so either everything he says is false, or everything is true). Which is it? (c) What, if anything, can the waiter conclude about the ingredients in Geoﬀ’s desired calzone? 6. Determine whether the following two statements are logically equivalent: ¬(P → Q) and P ∧ ¬Q. Explain how you know you are correct. 210 3. Symbolic Logic and Proofs 7. Are the statements P → (Q ∨ R) and (P → Q) ∨ (P → R) logically equivalent? 8. Simplify the following statements (so that negation only appears right before variables). (a) ¬(P → ¬Q). (b) (¬P ∨ ¬Q) → ¬(¬Q ∧ R). (c) ¬((P → ¬Q) ∨ ¬(R ∧ ¬R)). (d) It is false that if Sam is not a man then Chris is a woman, and that Chris is not a woman. 9. Use De Morgan’s Laws, and any other logical equivalence facts you know to simplify the following statements. Show all your steps. Your ﬁnal statements should have negations only appear directly next to the sentence variables or predicates (P, Q, E(x), etc.), and no double negations. It would be a good idea to use only conjunctions, disjunctions, and negations. (a) ¬((¬P ∧ Q) ∨ ¬(R ∨ ¬S)). (b) ¬((¬P → ¬Q) ∧ (¬Q → R)) (careful with the implications). (c) For both parts above, verify your answers are correct using truth tables. That is, use a truth table to check that the given statement and your proposed simpliﬁcation are actually logically equivalent. 10. Consider the statement, “If a number is triangular or square, then it is not prime” (a) Make a truth table for the statement (T ∨ S) → ¬P. (b) If you believed the statement was false, what properties would a counterexample need to possess? Explain by referencing your truth table. (c) If the statement were true, what could you conclude about the number 5657, which is deﬁnitely prime? Again, explain using the truth table. 11. Tommy Flanagan was telling you what he ate yesterday afternoon. He tells you, “I had either popcorn or raisins. Also, if I had cucumber sandwiches, then I had soda. But I didn’t drink soda or tea.” Of 3.1. Propositional Logic 211 course you know that Tommy is the worlds worst liar, and everything he says is false. What did Tommy eat? Justify your answer by writing all of Tommy’s statements using sentence variables (P, Q, R, S, T), taking their negations, and using these to deduce what Tommy actually ate. 12. Determine if the following deduction rule is valid: P∨Q ¬P ∴ Q 13. Determine if the following is a valid deduction rule: P → (Q ∨ R) ¬(P → Q) ∴ R 14. Determine if the following is a valid deduction rule: (P ∧ Q) → R ¬P ∨ ¬Q ∴ ¬R 15. Can you chain implications together? That is, if P → Q and Q → R, does that means the P → R? Can you chain more implications together? Let’s ﬁnd out: (a) Prove that the following is a valid deduction rule: ∴ P→Q Q→R P→R (b) Prove that the following is a valid deduction rule for any n ≥ 2: P1 → P2 P2 → P3 .. . ∴ Pn−1 → Pn P1 → Pn . I suggest you don’t go through the trouble of writing out a 2n row truth table. Instead, you should use part (a) and mathematical induction. 16. We can also simplify statements in predicate logic using our rules for passing negations over quantiﬁers, and then applying propositional logical equivalence to the “inside” propositional part. Simplify the 212 3. Symbolic Logic and Proofs statements below (so negation appears only directly next to predicates). (a) ¬∃x∀y(¬O(x) ∨ E(y)). (b) ¬∀x¬∀y¬(x < y ∧ ∃z(x < z ∨ y < z)). (c) There is a number n for which no other number is either less n than or equal to n. (d) It is false that for every number n there are two other numbers which n is between. 17. Simplify the statements below to the point that negation symbols occur only directly next to predicates. (a) ¬∀x∀y(x < y ∨ y < x). (b) ¬(∃xP(x) → ∀yP(y)). 18. Simplifying negations will be especially useful in the next section when we try to prove a statement by considering what would happen if it were false. For each statement below, write the negation of the statement as simply as possible. Don’t just say, “it is false that . . . ”. (a) Every number is either even or odd. (b) There is a sequence that is both arithmetic and geometric. (c) For all numbers n, if n is prime, then n + 3 is not prime. 19. Suppose P and Q are (possibly molecular) propositional statements. Prove that P and Q are logically equivalent if any only if P ↔ Q is a tautology. 20. Suppose P1 , P2 , . . . , Pn and Q are (possibly molecular) propositional statements. Suppose further that P1 P2 .. . ∴ Pn Q is a valid deduction rule. Prove that the statement (P1 ∧ P2 ∧ · · · ∧ Pn ) → Q is a tautology. 3.2. Proofs 3.2 Proofs Investigate! Decide which of the following are valid proofs of the following statement: If ab is an even number, then a or b is even. 1. Suppose a and b are odd. That is, a 2k + 1 and b 2m + 1 for some integers k and m. Then ab (2k + 1)(2m + 1) 4km + 2k + 2m + 1 2(2km + k + m) + 1. Therefore ab is odd. 2. Assume that a or b is even - say it is a (the case where b is even is identical). That is, a 2k for some integer k. Then ab (2k)b 2(kb). Thus ab is even. 3. Suppose that ab is even but a and b are both odd. Namely, ab 2n, a 2k + 1 and b 2j + 1 for some integers n, k, and j. Then 2n (2k + 1)(2 j + 1) 2n 4k j + 2k + 2 j + 1 n 2k j + k + j + 0.5. But since 2k j + k + j is an integer, this says that the integer n is equal to a non-integer, which is impossible. 4. Let ab be an even number, say ab 2n, and a be an odd number, say a 2k + 1. ab (2k + 1)b 2n 2kb + b 2n − 2kb b 2(n − kb) b. ! Therefore b must be even. Attempt the above activity before proceeding ! 213 214 3. Symbolic Logic and Proofs Anyone who doesn’t believe there is creativity in mathematics clearly has not tried to write proofs. Finding a way to convince the world that a particular statement is necessarily true is a mighty undertaking and can often be quite challenging. There is not a guaranteed path to success in the search for proofs. For example, in the summer of 1742, a German mathematician by the name of Christian Goldbach wondered whether every even integer greater than 2 could be written as the sum of two primes. Centuries later, we still don’t have a proof of this apparent fact (computers have checked that “Goldbach’s Conjecture” holds for all numbers less than 4 × 1018 , which leaves only inﬁnitely many more numbers to check). Writing proofs is a bit of an art. Like any art, to be truly great at it, you need some sort of inspiration, as well as some foundational technique. Just as musicians can learn proper ﬁngering, and painters can learn the proper way to hold a brush, we can look at the proper way to construct arguments. A good place to start might be to study a classic. Theorem 3.2.1 There are inﬁnitely many primes. Proof. Suppose this were not the case. That is, suppose there are only ﬁnitely many primes. Then there must be a last, largest prime, call it p. Consider the number N p! + 1 (p · (p − 1) · · · · 3 · 2 · 1) + 1. Now N is certainly larger than p. Also, N is not divisible by any number less than or equal to p, since every number less than or equal to p divides p!. Thus the prime factorization of N contains prime numbers (possibly just N itself) all greater than p. So p is not the largest prime, a contradiction. Therefore there are inﬁnitely many primes. qed This proof is an example of a proof by contradiction, one of the standard styles of mathematical proof. First and foremost, the proof is an argument. It contains sequence of statements, the last being the conclusion which follows from the previous statements. The argument is valid so the conclusion must be true if the premises are true. Let’s go through the proof line by line. 1. Suppose there are only ﬁnitely many primes. [this is a premise. Note the use of “suppose.”] 2. There must be a largest prime, call it p. [follows from line 1, by the deﬁnition of “ﬁnitely many.”] 3. Let N p! + 1. [basically just notation, although this is the inspired part of the proof; looking at p! + 1 is the key insight.] 4. N is larger than p. [by the deﬁnition of p!] 3.2. Proofs 215 5. N is not divisible by any number less than or equal to p. [by deﬁnition, p! is divisible by each number less than or equal to p, so p! + 1 is not.] 6. The prime factorization of N contains prime numbers greater than p. [since N is divisible by each prime number in the prime factorization of N, and by line 5.] 7. Therefore p is not the largest prime. [by line 6, N is divisible by a prime larger than p.] 8. This is a contradiction. [from line 2 and line 7: the largest prime is p and there is a prime larger than p.] 9. Therefore there are inﬁnitely many primes. [from line 1 and line 8: our only premise lead to a contradiction, so the premise is false.] We should say a bit more about the last line. Up through line 8, we have a valid argument with the premise “there are only ﬁnitely many primes” and the conclusion “there is a prime larger than the largest prime.” This is a valid argument as each line follows from previous lines. So if the premises are true, then the conclusion must be true. However, the conclusion is NOT true. The only way out: the premise must be false. The sort of line-by-line analysis we did above is a great way to really understand what is going on. Whenever you come across a proof in a textbook, you really should make sure you understand what each line is saying and why it is true. Additionally, it is equally important to understand the overall structure of the proof. This is where using tools from logic is helpful. Luckily there are a relatively small number of standard proof styles that keep showing up again and again. Being familiar with these can help understand proof, as well as give ideas of how to write your own. Direct Proof The simplest (from a logic perspective) style of proof is a direct proof. Often all that is required to prove something is a systematic explanation of what everything means. Direct proofs are especially useful when proving implications. The general format to prove P → Q is this: Assume P. Explain, explain, . . . , explain. Therefore Q. Often we want to prove universal statements, perhaps of the form ∀x(P(x) → Q(x)). Again, we will want to assume P(x) is true and deduce Q(x). But what about the x? We want this to work for all x. We accomplish this by ﬁxing x to be an arbitrary element (of the sort we are interested in). Here are a few examples. First, we will set up the proof structure for a direct proof, then ﬁll in the details. 216 3. Symbolic Logic and Proofs Example 3.2.2 Prove: For all integers n, if n is even, then n 2 is even. Solution. The format of the proof will be this: Let n be an arbitrary integer. Assume that n is even. Explain explain explain. Therefore n 2 is even. To ﬁll in the details, we will basically just explain what it means for n to be even, and then see what that means for n 2 . Here is a complete proof. Proof. Let n be an arbitrary integer. Suppose n is even. Then n 2k for some integer k. Now n 2 (2k)2 4k 2 2(2k 2 ). Since 2k 2 is an integer, n 2 is even. qed Example 3.2.3 Prove: For all integers a, b, and c, if a|b and b|c then a|c. (Here x| y, read “x divides y” means that y is a multiple of x, i.e., that x will divide into y without remainder). Solution. Even before we know what the divides symbol means, we can set up a direct proof for this statement. It will go something like this: Let a, b, and c be arbitrary integers. Assume that a|b and b|c. Dot dot dot. Therefore a|c. How do we connect the dots? We say what our hypothesis (a|b and b|c) really means and why this gives us what the conclusion (a|c) really means. Another way to say that a|b is to say that b ka for some integer k (that is, that b is a multiple of a). What are we going for? That c la, for some integer l (because we want c to be a multiple of a). Here is the complete proof. Proof. Let a, b, and c be integers. Assume that a|b and b|c. In other words, b is a multiple of a and c is a multiple of b. So there are integers k and j such that b ka and c jb. Combining these (through substitution) we get that c jka. But jk is an integer, so this says that c is a multiple of a. Therefore a|c. qed Proof by Contrapositive Recall that an implication P → Q is logically equivalent to its contrapositive ¬Q → ¬P. There are plenty of examples of statements which are hard to prove directly, but whose contrapositive can easily be proved directly. This is all that proof by contrapositive does. It gives a direct 3.2. Proofs 217 proof of the contrapositive of the implication. This is enough because the contrapositive is logically equivalent to the original implication. The skeleton of the proof of P → Q by contrapositive will always look roughly like this: Assume ¬Q. Explain, explain, . . . explain. Therefore ¬P. As before, if there are variables and quantiﬁers, we set them to be arbitrary elements of our domain. Here are two examples: Example 3.2.4 Is the statement “for all integers n, if n 2 is even, then n is even” true? Solution. This is the converse of the statement we proved above using a direct proof. From trying a few examples, this statement deﬁnitely appears to be true. So let’s prove it. A direct proof of this statement would require ﬁxing an arbitrary n and assuming that n 2 is even. But it is not at all clear how this would allow us to conclude anything about n. Just because n 2 2k does not in itself suggest how we could write n as a multiple of 2. Try something else: write the contrapositive of the statement. We get, for all integers n, if n is odd then n 2 is odd. This looks much more promising. Our proof will look something like this: Let n be an arbitrary integer. Suppose that n is not even. This means that . . . . In other words . . . . But this is the same as saying . . . . Therefore n 2 is not even. Now we ﬁll in the details: Proof. We will prove the contrapositive. Let n be an arbitrary integer. Suppose that n is not even, and thus odd. Then n 2k + 1 for some integer k. Now n 2 (2k +1)2 4k 2 +4k +1 2(2k 2 +2k)+1. Since 2k 2 + 2k is an integer, we see that n 2 is odd and therefore not even. qed Example 3.2.5 Prove: for all integers a and b, if a + b is odd, then a is odd or b is odd. Solution. The problem with trying a direct proof is that it will be hard to separate a and b from knowing something about a + b. On the other hand, if we know something about a and b separately, then combining them might give us information about a + b. The 218 3. Symbolic Logic and Proofs contrapositive of the statement we are trying to prove is: for all integers a and b, if a and b are even, then a + b is even. Thus our proof will have the following format: Let a and b be integers. Assume that a and b are both even. la la la. Therefore a + b is even. Here is a complete proof: Proof. Let a and b be integers. Assume that a and b are even. Then a 2k and b 2l for some integers k and l. Now a + b 2k + 2l 2(k +1). Since k + l is an integer, we see that a + b is even, completing the proof. qed Note that our assumption that a and b are even is really the negation of a or b is odd. We used De Morgan’s law here. We have seen how to prove some statements in the form of implications: either directly or by contrapositive. Some statements are not written as implications to begin with. Example 3.2.6 Consider the following statement: for every prime number p, either p 2 or p is odd. We can rephrase this: for every prime number p, if p , 2, then p is odd. Now try to prove it. Solution. Proof. Let p be an arbitrary prime number. Assume p is not odd. So p is divisible by 2. Since p is prime, it must have exactly two divisors, and it has 2 as a divisor, so p must be divisible by only 1 and 2. Therefore p 2. This completes the proof (by contrapositive). qed Proof by Contradiction There might be statements which really cannot be rephrased as implica√ tions. For example, “ 2 is irrational.” In this case, it is hard to know where to start. What can we assume? Well, say we want to prove the statement P. What if we could prove that ¬P → Q where Q was false? If this implication is true, and Q is false, what can we say about ¬P? It must be false as well, which makes P true! This is why proof by contradiction works. If we can prove that ¬P leads to a contradiction, then the only conclusion is that ¬P is false, so P is true. That’s what we wanted to prove. In other words, if it is impossible for P to be false, P must be true. Here are three examples of proofs by contradiction: 3.2. Proofs 219 Example 3.2.7 √ Prove that 2 is irrational. Solution. √ Proof. Suppose not. Then 2 is equal to a fraction ba . Without loss of generality, assume ba is in lowest terms (otherwise reduce the fraction). So, a2 2 2 b 2b 2 a 2 . Thus a 2 is even, and as such a is even. So a 2k for some integer k, and a 2 4k 2 . We then have, 2b 2 4k 2 b 2 2k 2 . Thus b 2 is even, and as such b is even. Since a is also even, √ we see that ba is not in lowest terms, a contradiction. Thus 2 is irrational. qed Example 3.2.8 Prove: There are no integers x and y such that x 2 4y + 2. Solution. Proof. We proceed by contradiction. So suppose there are integers x and y such that x 2 4y + 2 2(2y + 1). So x 2 is even. We have seen that this implies that x is even. So x 2k for some integer k. Then x 2 4k 2 . This in turn gives 2k 2 (2y + 1). But 2k 2 is even, and 2y + 1 is odd, so these cannot be equal. Thus we have a contradiction, so there must not be any integers x and y such that x 2 4y + 2. qed Example 3.2.9 The Pigeonhole Principle: If more than n pigeons ﬂy into n pigeon holes, then at least one pigeon hole will contain at least two pigeons. Prove this! 220 3. Symbolic Logic and Proofs Solution. Proof. Suppose, contrary to stipulation, that each of the pigeon holes contain at most one pigeon. Then at most, there will be n pigeons. But we assumed that there are more than n pigeons, so this is impossible. Thus there must be a pigeonhole with more than one pigeon. qed While we phrased this proof as a proof by contradiction, we could have also used a proof by contrapositive since our contradiction was simply the negation of the hypothesis. Sometimes this will happen, in which case you can use either style of proof. There are examples however where the contradiction occurs “far away” from the original statement. Proof by (counter) Example It is almost NEVER okay to prove a statement with just an example. Certainly none of the statements proved above can be proved through an example. This is because in each of those cases we are trying to prove that something holds of all integers. We claim that n 2 being even implies that n is even, no matter what integer n we pick. Showing that this works for n 4 is not even close to enough. This cannot be stressed enough. If you are trying to prove a statement of the form ∀xP(x), you absolutely CANNOT prove this with an example.1 However, existential statements can be proven this way. If we want to prove that there is an integer n such that n 2 − n + 41 is not prime, all we need to do is ﬁnd one. This might seem like a silly thing to want to prove until you try a few values for n. n2 n − n + 41 1 41 2 43 3 47 4 53 5 61 6 71 7 83 So far we have gotten only primes. You might be tempted to conjecture, “For all positive integers n, the number n 2 − n + 41 is prime.” If you wanted to prove this, you would need to use a direct proof, a proof by contrapositive, or another style of proof, but certainly it is not enough to give even 7 examples. In fact, we can prove this conjecture is false by proving its negation: “There is a positive integer n such that n 2 − n + 41 is not prime.” Since this is an existential statement, it suﬃces to show that there does indeed exist such a number. In fact, we can quickly see that n 41 will give 412 which is certainly not prime. You might say that this is a counterexample to the conjecture 1This is not to say that looking at examples is a waste of time. Doing so will often give you an idea of how to write a proof. But the examples do not belong in the proof. 3.2. Proofs 221 that n 2 − n + 41 is always prime. Since so many statements in mathematics are universal, making their negations existential, we can often prove that a statement is false (if it is) by providing a counterexample. Example 3.2.10 Above we proved, “for all integers a and b, if a + b is odd, then a is odd or b is odd.” Is the converse true? Solution. The converse is the statement, “for all integers a and b, if a is odd or b is odd, then a + b is odd.” This is false! How do we prove it is false? We need to prove the negation of the converse. Let’s look at the symbols. The converse is ∀a∀b((O(a) ∨ O(b)) → O(a + b)). We want to prove the negation: ¬∀a∀b((O(a) ∨ O(b)) → O(a + b)). Simplify using the rules from the previous sections: ∃a∃b((O(a) ∨ O(b)) ∧ ¬O(a + b)). As the negation passed by the quantiﬁers, they changed from ∀ to ∃. We then needed to take the negation of an implication, which is equivalent to asserting the if part and not the then part. Now we know what to do. To prove that the converse is false we need to ﬁnd two integers a and b so that a is odd or b is odd, but a + b is not odd (so even). That’s easy: 1 and 3. (remember, “or” means one or the other or both). Both of these are odd, but 1 + 3 4 is not odd. Proof by Cases We could go on and on and on about diﬀerent proof styles (we haven’t even mentioned induction or combinatorial proofs here), but instead we will end with one ﬁnal useful technique: proof by cases. The idea is to prove that P is true by proving that Q → P and ¬Q → P for some statement Q. So no matter what, whether or not Q is true, we know that P is true. In fact, we could generalize this. Suppose we want to prove P. We know that at least one of the statements Q 1 , Q2 , . . . , Q n is true. If we can show that Q 1 → P and Q2 → P and so on all the way to Q n → P, 222 3. Symbolic Logic and Proofs then we can conclude P. The key thing is that we want to be sure that one of our cases (the Q i ’s) must be true no matter what. If that last paragraph was confusing, perhaps an example will make things better. Example 3.2.11 Prove: For any integer n, the number (n 3 − n) is even. Solution. It is hard to know where to start this, because we don’t know much of anything about n. We might be able to prove that n 3 −n is even if we knew that n was even. In fact, we could probably prove that n 3 − n was even if n was odd. But since n must either be even or odd, this will be enough. Here’s the proof. Proof. We consider two cases: if n is even or if n is odd. Case 1: n is even. Then n 2k for some integer k. This give n 3 − n 8k 3 − 2k 2(4k 2 − k), and since 4k 2 − k is an integer, this says that n 3 − n is even. Case 2: n is odd. Then n 2k + 1 for some integer k. This gives n 3 − n (2k + 1)3 − (2k + 1) 8k 3 + 6k 2 + 6k + 1 − 2k − 1 2(4k 3 + 3k 2 + 2k), and since 4k 3 + 3k 2 + 2k is an integer, we see that n 3 − n is even again. Since n 3 − n is even in both exhaustive cases, we see that n 3 − n is indeed always even. qed 3.2. Proofs 223 Exercises 1. Consider the statement “for all integers a and b, if a + b is even, then a and b are even” (a) Write the contrapositive of the statement. (b) Write the converse of the statement. (c) Write the negation of the statement. (d) Is the original statement true or false? Prove your answer. (e) Is the contrapositive of the original statement true or false? Prove your answer. (f) Is the converse of the original statement true or false? Prove your answer. (g) Is the negation of the original statement true or false? Prove your answer. 2. For each of the statements below, say what method of proof you should use to prove them. Then say how the proof starts and how it ends. Bonus points for ﬁlling in the middle. (a) There are no integers x and y such that x is a prime greater than 5 and x 6y + 3. (b) For all integers n, if n is a multiple of 3, then n can be written as the sum of consecutive integers. (c) For all integers a and b, if a 2 + b 2 is odd, then a or b is odd. 3. Consider the statement: for all integers n, if n is even then 8n is even. (a) Prove the statement. What sort of proof are you using? (b) Is the converse true? Prove or disprove. 4. The game TENZI comes with 40 six-sided dice (each numbered 1 to 6). Suppose you roll all 40 dice. (a) Prove that there will be at least seven dice that land on the same number. (b) How many dice would you have to roll before you were guaranteed that some four of them would all match or all be diﬀerent? Prove your answer. 5. Prove that for all integers n, it is the case that n is even if and only if 3n is even. That is, prove both implications: if n is even, then 3n is even, and if 3n is even, then n is even. 224 6. 7. 3. Symbolic Logic and Proofs √ Prove that 3 is irrational. Consider the statement: for all integers a and b, if a is even and b is a multiple of 3, then ab is a multiple of 6. (a) Prove the statement. What sort of proof are you using? (b) State the converse. Is it true? Prove or disprove. 8. Prove the statement: For all integers n, if 5n is odd, then n is odd. Clearly state the style of proof you are using. 9. Prove the statement: For all integers a, b, and c, if a 2 + b 2 c 2 , then a or b is even. 10. Suppose that you would like to prove the following implication: For all numbers n, if n is prime then n is solitary. Write out the beginning and end of the argument if you were to prove the statement, (a) Directly (b) By contrapositive (c) By contradiction You do not need to provide details for the proofs (since you do not know what solitary means). However, make sure that you provide the ﬁrst few and last few lines of the proofs so that we can see that logical structure you would follow. 11. Suppose you have a collection of 5-cent stamps and 8-cent stamps. We saw earlier that it is possible to make any amount of postage greater than 27 cents using combinations of both these types of stamps. But, let’s ask some other questions: (a) Prove that if you only use an even number of both types of stamps, the amount of postage you make must be even. (b) Suppose you made an even amount of postage. Prove that you used an even number of at least one of the types of stamps. (c) Suppose you made exactly 72 cents of postage. Prove that you used at least 6 of one type of stamp. (x + y)2 . Note, you will need to 4 prove two “directions” here: the “if” and the “only if” part. 12. Prove: x y if and only if x y 13. Prove that log(7) is irrational. 3.2. Proofs 225 14. Prove that there are no integer solutions to the equation x 2 4y + 3. 15. Prove that every prime number greater than 3 is either one more or one less than a multiple of 6. 16. Your “friend” has shown you a “proof” he wrote to show that 1 3. Here is the proof: Proof. I claim that 1 3. Of course we can do anything to one side of an equation as long as we also do it to the other side. So subtract 2 from both sides. This gives −1 1. Now square both sides, to get 1 1. And we all agree this is true. qed What is going on here? Is your friend’s argument valid? Is the argument a proof of the claim 1 3? Carefully explain using what we know about logic. 17. A standard deck of 52 cards consists of 4 suites (hearts, diamonds, spades and clubs) each containing 13 diﬀerent values (Ace, 2, 3, . . . , 10, J, Q, K). If you draw some number of cards at random you might or might not have a pair (two cards with the same value) or three cards all of the same suit. However, if you draw enough cards, you will be guaranteed to have these. For each of the following, ﬁnd the smallest number of cards you would need to draw to be guaranteed having the speciﬁed cards. Prove your answers. (a) Three of a kind (for example, three 7’s). (b) A ﬂush of ﬁve cards (for example, ﬁve hearts). (c) Three cards that are either all the same suit or all diﬀerent suits. 18. Suppose you are at a party with 19 of your closest friends (so including you, there are 20 people there). Explain why there must be least two people at the party who are friends with the same number of people at the party. Assume friendship is always reciprocated. 19. Your friend has given you his list of 115 best Doctor Who episodes (in order of greatness). It turns out that you have seen 60 of them. Prove that there are at least two episodes you have seen that are exactly four episodes apart on your friend’s list. 20. Suppose you have an n × n chessboard but your dog has eaten one of the corner squares. Can you still cover the remaining squares with dominoes? What needs to be true about n? Give necessary and suﬃcient conditions (that is, say exactly which values of n work and which do not work). Prove your answers. 226 3. Symbolic Logic and Proofs 21. What if your n × n chessboard is missing two opposite corners? Prove that no matter what n is, you will not be able to cover the remaining squares with dominoes. 3.3. Chapter Summary 3.3 227 Chapter Summary We have considered logic both as its own sub-discipline of mathematics, and as a means to help us better understand and write proofs. In either view, we noticed that mathematical statements have a particular logical form, and analyzing that form can help make sense of the statement. At the most basic level, a statement might combine simpler statements using logical connectives. We often make use of variables, and quantify over those variables. How to resolve the truth or falsity of a statement based on these connectives and quantiﬁers is what logic is all about. From this, we can decide whether two statements are logically equivalent or if one or more statements (logically) imply another. When writing proofs (in any area of mathematics) our goal is to explain why a mathematical statement is true. Thus it is vital that our argument implies the truth of the statement. To be sure of this, we ﬁrst must know what it means for the statement to be true, as well as ensure that the statements that make up the proof correctly imply the conclusion. A ﬁrm understanding of logic is required to check whether a proof is correct. There is, however, another reason that understanding logic can be helpful. Understanding the logical structure of a statement often gives clues as how to write a proof of the statement. This is not to say that writing proofs is always straight forward. Consider again the Goldbach conjecture: Every even number greater than 2 can be written as the sum of two primes. We are not going to try to prove the statement here, but we can at least say what a proof might look like, based on the logical form of the statement. Perhaps we should write the statement in an equivalent way which better highlights the quantiﬁers and connectives: For all integers n, if n is even and greater than 2, then there exists integers p and q such that p and q are prime and n p+q. What would a direct proof look like? Since the statement starts with a universal quantiﬁer, we would start by, ``Let n be an arbitrary integer." The rest of the statement is an implication. In a direct proof we assume the “if” part, so the next line would be, “Assume n is greater than 2 and is even.” I have no idea what comes next, but eventually, we would need to ﬁnd two prime numbers p and q (depending on n) and explain how we know that n p + q. Or maybe we try a proof by contradiction. To do this, we ﬁrst assume the negation of the statement we want to prove. What is the negation? From what we have studied we should be able to see that it is, 228 3. Symbolic Logic and Proofs There is an integer n such that n is even and greater than 2, but for all integers p and q, either p or q is not prime or n , p + q. Could this statement be true? A proof by contradiction would start by assuming it was and eventually conclude with a contradiction, proving that our assumption of truth was incorrect. And if you can ﬁnd such a contradiction, you will have proved the most famous open problem in mathematics. Good luck. Chapter Review 1. Complete a truth table for the statement ¬P → (Q ∧ R). 2. Suppose you know that the statement “if Peter is not tall, then Quincy is fat and Robert is skinny” is false. What, if anything, can you conclude about Peter and Robert if you know that Quincy is indeed fat? Explain (you may reference problem 3.3.1). 3. Are the statements P → (Q ∨ R) and (P → Q) ∨ (P → R) logically equivalent? Explain your answer. 4. Is the following a valid deduction rule? Explain. P→Q P→R ∴ P → (Q ∧ R). 5. Write the negation, converse and contrapositive for each of the statements below. (a) If the power goes oﬀ, then the food will spoil. (b) If the door is closed, then the light is oﬀ. (c) ∀x(x < 1 → x 2 < 1). (d) For all natural numbers n, if n is prime, then n is solitary. (e) For all functions f , if f is diﬀerentiable, then f is continuous. (f) For all integers a and b, if a · b is even, then a and b are even. (g) For every integer x and every integer y there is an integer n such that if x > 0 then nx > y. (h) For all real numbers x and y, if x y 0 then x 0 or y 0. (i) For every student in Math 228, if they do not understand implications, then they will fail the exam. 3.3. Chapter Summary 6. 229 Consider the statement: for all integers n, if n is even and n ≤ 7 then n is negative or n ∈ {0, 2, 4, 6}. (a) Is the statement true? Explain why. (b) Write the negation of the statement. Is it true? Explain. (c) State the contrapositive of the statement. Is it true? Explain. (d) State the converse of the statement. Is it true? Explain. 7. Consider the statement: ∀x(∀y(x + y y) → ∀z(x · z 0)). (a) Explain what the statement says in words. Is this statement true? Be sure to state what you are taking the universe of discourse to be. (b) Write the converse of the statement, both in words and in symbols. Is the converse true? (c) Write the contrapositive of the statement, both in words and in symbols. Is the contrapositive true? (d) Write the negation of the statement, both in words and in symbols. Is the negation true? 8. Simplify the following. (a) ¬(¬(P ∧ ¬Q) → ¬(¬R ∨ ¬(P → R))). (b) ¬∃x¬∀y¬∃z(z x + y → ∃w(x − y w)). 9. Consider the statement: for all integers n, if n is odd, then 7n is odd. (a) Prove the statement. What sort of proof are you using? (b) Prove the converse. What sort of proof are you using? 10. Suppose you break your piggy bank and scoop up a handful of 22 coins (pennies, nickels, dimes and quarters). (a) Prove that you must have at least 6 coins of a single denomination. (b) Suppose you have an odd number of pennies. Prove that you must have an odd number of at least one of the other types of coins. (c) How many coins would you need to scoop up to be sure that you either had 4 coins that were all the same or 4 coins that were all diﬀerent? Prove your answer. 230 3. Symbolic Logic and Proofs 11. You come across four trolls playing bridge. They declare: Troll 1: All trolls here see at least one knave. Troll 2: I see at least one troll that sees only knaves. Troll 3: Some trolls are scared of goats. Troll 4: All trolls are scared of goats. Are there any trolls that are not scared of goats? Recall, of course, that all trolls are either knights (who always tell the truth) or knaves (who always lie). Chapter 4 Graph Theory Investigate! In the time of Euler, in the town of Königsberg in Prussia, there was a river containing two islands. The islands were connected to the banks of the river by seven bridges (as seen below). The bridges were very beautiful, and on their days oﬀ, townspeople would spend time walking over the bridges. As time passed, a question arose: was it possible to plan a walk so that you cross each bridge once and only once? Euler was able to answer this question. Are you? ! Attempt the above activity before proceeding ! Graph Theory is a relatively new area of mathematics, ﬁrst studied by the super famous mathematician Leonhard Euler in 1735. Since then it has blossomed in to a powerful tool used in nearly every branch of science and is currently an active area of mathematics research. The problem above, known as the Seven Bridges of Königsberg, is the problem that originally inspired graph theory. Consider a “diﬀerent” problem: Below is a drawing of four dots connected by some lines. Is it possible to trace over each line once and only once (without lifting up your pencil, starting and ending on a dot)? There is an obvious connection between these two problems. Any path in the dot and line drawing corresponds exactly to a path over the bridges of Königsberg. 231 232 4. Graph Theory Pictures like the dot and line drawing are called graphs. Graphs are made up of a collection of dots called vertices and lines connecting those dots called edges. When two vertices are connected by an edge, we say they are adjacent. The nice thing about looking at graphs instead of pictures of rivers, islands and bridges is that we now have a mathematical object to study. We have distilled the “important” parts of the bridge picture for the purposes of the problem. It does not matter how big the islands are, what the bridges are made out of, if the river contains alligators, etc. All that matters is which land masses are connected to which other land masses, and how many times. We will return to the question of ﬁnding paths through graphs later. But ﬁrst, here are a few other situations you can represent with graphs: Example 4.0.1 Al, Bob, Cam, Dan, and Euler are all members of the social networking website Facebook. The site allows members to be “friends” with each other. It turns out that Al and Cam are friends, as are Bob and Dan. Euler is friends with everyone. Represent this situation with a graph. Solution. Each person will be represented by a vertex and each friendship will be represented by an edge. That is, two vertices will be adjacent (there will be an edge between them) if and only if the people represented by those vertices are friends. A C B E D Example 4.0.2 Each of three houses must be connected to each of three utilities. Is it possible to do this without any of the utility lines crossing? Solution. We will answer this question later. For now, notice how we would ask this question in the context of graph theory. We are really asking whether it is possible to redraw the graph below without any edges crossing (except at vertices). Think of the top row as the houses, bottom row as the utilities. 4.1. Definitions 4.1 233 Definitions Investigate! Which (if any) of the graphs below are the same? The graphs above are unlabeled. Usually we think of a graph as having a speciﬁc set of vertices. Which (if any) of the graphs below are the same? b d f b c f c b f a c e a d e a e d v1 v2 v3 v6 v5 v4 Actually, all the graphs we have seen above are just drawings of graphs. A graph is really an abstract mathematical object consisting of two sets V and E where E is a set of 2-element subsets of V. Are the graphs below the same or diﬀerent? Graph 1: V {a, b, c, d, e}, E {{a, b}, {a, c}, {a, d}, {a, e}, {b, c}, {d, e}}. Graph 2: V {v 1 , v 2 , v 3 , v4 , v 5 }, E {{v1 , v 3 }, {v 1 , v 5 }, {v 2 , v 4 }, {v2 , v 5 }, {v 3 , v 5 }, {v 4 , v 5 }}. ! Attempt the above activity before proceeding ! Before we start studying graphs, we need to agree upon what a graph is. While we almost always think of graphs as pictures (dots connected by lines) this is fairly ambiguous. Do the lines need to be straight? Does it matter how long the lines are or how large the dots are? Can there be two lines connecting the same pair of dots? Can one line connect three dots? The way we avoid ambiguities in mathematics is to provide concrete and rigorous deﬁnitions. Crafting good deﬁnitions is not easy, but it is incredibly important. The deﬁnition is the agreed upon starting point from which all truths in mathematics proceed. Is there a graph with no edges? We have to look at the deﬁnition to see if this is possible. We want our deﬁnition to be precise and unambiguous, but it also must agree with our intuition for the objects we are studying. It needs to be useful: we could deﬁne a graph to be a six legged mammal, but that 234 4. Graph Theory would not let us solve any problems about bridges. Instead, here is the (now) standard deﬁnition of a graph. Graph Deﬁnition. A graph is an ordered pair G (V, E) consisting of a nonempty set V (called the vertices) and a set E (called the edges) of two-element subsets of V. Strange. Nowhere in the deﬁnition is there talk of dots or lines. From the deﬁnition, a graph could be ({a, b, c, d}, {{a, b}, {a, c}, {b, c}, {b, d}, {c, d}}). Here we have a graph with four vertices (the letters a, b, c, d) and ﬁve edges (the pairs {a, b}, {a, c}, {b, c}, {b, d}, {c, d})). Looking at sets and sets of 2-element sets is diﬃcult to process. That is why we often draw a representation of these sets. We put a dot down for each vertex, and connect two dots with a line precisely when those two vertices are one of the 2-element subsets in our set of edges. Thus one way to draw the graph described above is this: a b c d However we could also have drawn the graph diﬀerently. For example either of these: a d c b a b c d We should be careful about what it means for two graphs to be “the same.” Actually, given our deﬁnition, this is easy: Are the vertex sets equal? Are the edge sets equal? We know what it means for sets to be equal, and graphs are nothing but a pair of two special sorts of sets. Example 4.1.1 Are the graphs below equal? G1 ({a, b, c}, {{a, b}, {b, c}}); G2 ({a, b, c}, {{a, c}, {c, b}}). Solution. No. Here the vertex sets of each graph are equal, which is a good start. Also, both graphs have two edges. In the ﬁrst graph, we have edges {a, b} and {b, c}, while in the second graph we have 4.1. Definitions 235 edges {a, c} and {c, b}. Now we do have {b, c} {c, b}, so that is not the problem. The issue is that {a, b} , {a, c}. Since the edge sets of the two graphs are not equal (as sets), the graphs are not equal (as graphs). Even if two graphs are not equal, they might be basically the same. The graphs in the previous example could be drawn like this: G1 G2 a c b a c b Graphs that are basically the same (but perhaps not equal) are called isomorphic. We will give a precise deﬁnition of this term after a quick example: Example 4.1.2 Consider the graphs: G1 {V1 , E1 } where V1 {a, b, c} and E1 {{a, b}, {a, c}, {b, c}}; G2 {V2 , E2 } where V2 {u, v, w} and E2 {{u, v}, {u, w}, {v, w}}. Are these graphs the same? Solution. The two graphs are NOT equal. It is enough to notice that V1 , V2 since a ∈ V1 but a < V2 . However, both of these graphs consist of three vertices with edges connecting every pair of vertices. We can draw them as follows: a b u c v w Clearly we want to say these graphs are basically the same, so while they are not equal, they will be isomorphic. We can rename the vertices of one graph and get the second graph as the result. Intuitively, graphs are isomorphic if they are basically the same, or better yet, if they are the same except for the names of the vertices. To make the concept of renaming vertices precise, we give the following deﬁnitions: 236 4. Graph Theory Isomorphic Graphs. An isomorphism between two graphs G1 and G2 is a bĳection f : V1 → V2 between the vertices of the graphs such that {a, b} is an edge in G1 if and only if { f (a), f (b)} is an edge in G2 . Two graphs are isomorphic if there is an isomorphism between them. In this case we write G1 G2 . An isomorphism is simply a function which renames the vertices. It must be a bĳection so every vertex gets a new name. These newly named vertices must be connected by edges precisely when they were connected by edges with their old names. Example 4.1.3 Decide whether the graphs G1 {V1 , E1 } and G2 {V2 , E2 } are equal or isomorphic. V1 {a, b, c, d}, E1 {{a, b}, {a, c}, {a, d}, {c, d}} V2 {a, b, c, d}, E2 {{a, b}, {a, c}, {b, c}, {c, d}} Solution. The graphs are NOT equal, since {a, d} ∈ E1 but {a, d} < E2 . However, since both graphs contain the same number of vertices and same number of edges, they might be isomorphic (this is not enough in most cases, but it is a good start). We can try to build an isomorphism. How about we say f (a) b, f (b) c, f (c) d and f (d) a. This is deﬁnitely a bĳection, but to make sure that the function is an isomorphism, we must make sure it respects the edge relation. In G1 , vertices a and b are connected by an edge. In G2 , f (a) b and f (b) c are connected by an edge. So far, so good, but we must check the other three edges. The edge {a, c} in G1 corresponds to { f (a), f (c)} {b, d}, but here we have a problem. There is no edge between b and d in G2 . Thus f is NOT an isomorphism. Not all hope is lost, however. Just because f is not an isomorphism does not mean that there is no isomorphism at all. We can try again. At this point it might be helpful to draw the graphs to see how they should match up. G1 : a d G2 : b c a d b c Alternatively, notice that in G1 , the vertex a is adjacent to every other vertex. In G2 , there is also a vertex with this property: c. So 4.1. Definitions 237 build the bĳection g : V1 → V2 by deﬁning g(a) c to start with. Next, where should we send b? In G1 , the vertex b is only adjacent to vertex a. There is exactly one vertex like this in G2 , namely d. So let g(b) d. As for the last two, in this example, we have a free choice: let g(c) b and g(d) a (switching these would be ﬁne as well). We should check that this really is an isomorphism. It is deﬁnitely a bĳection. We must make sure that the edges are respected. The four edges in G1 are {a, b}, {a, c}, {a, d}, {c, d}. Under the proposed isomorphism these become { g(a), g(b)}, { g(a), g(c)}, { g(a), g(d)}, { g(c), g(d)} {c, d}, {c, b}, {c, a}, {b, a}, which are precisely the edges in G2 . Thus g is an isomorphism, so G1 G2 Sometimes we will talk about a graph with a special name (like K n or the Peterson graph) or perhaps draw a graph without any labels. In this case we are really referring to all graphs isomorphic to any copy of that particular graph. A collection of isomorphic graphs is often called an isomorphism class. 1 There are other relationships between graphs that we care about, other than equality and being isomorphic. For example, compare the following pair of graphs: These are deﬁnitely not isomorphic, but notice that the graph on the right looks like it might be part of the graph on the left, especially if we draw it like this: 1This is not unlike geometry, where we might have more than one copy of a particular triangle. There instead of isomorphic we say congruent. 238 4. Graph Theory We would like to say that the smaller graph is a subgraph of the larger. We should give a careful deﬁnition of this. In fact, there are two reasonable notions for what a subgraph should mean. Subgraphs. We say that G′ (V ′ , E′) is a subgraph of G (V, E), and write G′ ⊆ G, provided V ′ ⊆ V and E′ ⊆ E. We say that G′ (V ′ , E′) is an induced subgraph of G (V, E) provided V ′ ⊆ V and every edge in E whose vertices are still in V ′ is also an edge in E′. Notice that every induced subgraph is also an ordinary subgraph, but not conversely. Think of a subgraph as the result of deleting some vertices and edges from the larger graph. For the subgraph to be an induced subgraph, we can still delete vertices, but now we only delete those edges that included the deleted vertices. Example 4.1.4 Consider the graphs: f f e d a b G1 d d c a f b G2 c a d b G3 c a b c G4 Here both G2 and G3 are subgraphs of G1 . But only G2 is an induced subgraph. Every edge in G1 that connects vertices in G2 is also an edge in G2 . In G3 , the edge {a, b} is in E1 but not E3 , even though vertices a and b are in V3 . The graph G4 is NOT a subgraph of G1 , even though it looks like all we did is remove vertex e. The reason is that in E4 we have the edge {c, f } but this is not an element of E1 , so we don’t have the required E4 ⊆ E1 . Back to some basic graph theory deﬁnitions. Notice that all the graphs we have drawn above have the property that no pair of vertices is connected more than once, and no vertex is connected to itself. Graphs like these are sometimes called simple, although we will just call them graphs. This is because our deﬁnition for a graph says that the edges form a set of 2-element subsets of the vertices. Remember that it doesn’t make sense to say a set contains an element more than once. So no pair of vertices can be connected by an edge more than once. Also, since each edge must be 4.1. Definitions 239 a set containing two vertices, we cannot have a single vertex connected to itself by an edge. That said, there are times we want to consider double (or more) edges and single edge loops. For example, the “graph” we drew for the Bridges of Königsberg problem had double edges because there really are two bridges connecting a particular island to the near shore. We will call these objects multigraphs. This is a good name: a multiset is a set in which we are allowed to include a single element multiple times. The graphs above are also connected: you can get from any vertex to any other vertex by following some path of edges. A graph that is not connected can be thought of as two separate graphs drawn close together. For example, the following graph is NOT connected because there is no path from a to b: a b Vertices in a graph do not always have edges between them. If we add all possible edges, then the resulting graph is called complete. That is, a graph is complete if every pair of vertices is connected by an edge. Since a graph is determined completely by which vertices are adjacent to which other vertices, there is only one complete graph with a given number of vertices. We give these a special name: K n is the complete graph on n vertices. Each vertex in K n is adjacent to n −1 other vertices. We call the number of edges emanating from a given vertex the degree of that vertex. So every vertex in K n has degree n − 1. How many edges does K n have? One might think the answer should be n(n − 1), since we count n − 1 edges n times (once for each vertex). However, each edge is incident to 2 vertices, so we counted every edge exactly twice. Thus there are n(n − 1)/2 edges in K n . Alternatively, we can say there are n2 edges, since to draw an edge we must choose 2 of the n vertices. In general, if we know the degrees of all the vertices in a graph, we can ﬁnd the number of edges. The sum of the degrees of all vertices will always be twice the number of edges, since each edge adds to the degree of two vertices. Notice this means that the sum of the degrees of all vertices in any graph must be even! This is our ﬁrst example of a general result about all graphs. It seems innocent enough, but we will use it to prove all sorts of other statements. So let’s give it a name and state it formally. 240 4. Graph Theory Lemma 4.1.5 Handshake Lemma. In any graph, the sum of the degrees of vertices in the graph is always twice the number of edges. The handshake lemma2 is sometimes called the degree sum formula, and can be written symbolically as Õ d(v) 2e. v∈V Here we are using the notation d(v) for the degree of the vertex v. One use for the lemma is to actually ﬁnd the number of edges in a graph. To do this, you must be given the degree sequence for the graph (or be able to ﬁnd it from other information). This is a list of every degree of every vertex in the graph, generally written in non-increasing order. Example 4.1.6 How many vertices and edges must a graph have if its degree sequence is (4, 4, 3, 3, 3, 2, 1)? Solution. The number of vertices is easy to ﬁnd: it is the number of degrees in the sequence: 7. To ﬁnd the number of edges, we compute the degree sum: 4 + 4 + 3 + 3 + 3 + 2 + 1 20, so the number of edges is half this: 10. The handshake lemma also tells use what is not possible. Example 4.1.7 At a recent math seminar, 9 mathematicians greeted each other by shaking hands. Is it possible that each mathematician shook hands with exactly 7 people at the seminar? Solution. It seems like this should be possible. Each mathematician chooses one person to not shake hands with. But this cannot happen. We are asking whether a graph with 9 vertices can have each vertex have degree 7. If such a graph existed, the sum of the degrees of the vertices would be 9 · 7 63. This would be twice the number of edges (handshakes) resulting in a graph with 31.5 edges. That is impossible. Thus at least one (in fact an odd number) of the 2A lemma is a mathematical statement that is primarily of importance in that it is used to establish other results. 4.1. Definitions 241 mathematicians must have shaken hands with an even number of people at the seminar. We can generalize the previous example to get the following proposition.3 Proposition 4.1.8 In any graph, the number of vertices with odd degree must be even. Proof. Suppose there were a graph with an odd number of vertices with odd degree. Then the sum of the degrees in the graph would be odd, which is impossible, by the handshake lemma. qed We will consider further applications of the handshake lemma in the exercises. One ﬁnal deﬁnition: we say a graph is bipartite if the vertices can be divided into two sets, A and B, with no two vertices in A adjacent and no two vertices in B adjacent. The vertices in A can be adjacent to some or all of the vertices in B. If each vertex in A is adjacent to all the vertices in B, then the graph is a complete bipartite graph, and gets a special name: K m,n , where |A| m and |B| n. The graph in the houses and utilities puzzle is K 3,3 . Named Graphs. Some graphs are used more than others, and get special names. Kn The complete graph on n vertices. K m,n The complete bipartite graph with sets of m and n vertices. Cn The cycle on n vertices, just one big loop. Pn The path on n + 1 vertices (so n edges), just one long path. K5 K 2,3 C6 P5 3A proposition is a general statement in mathematics, similar to a theorem, although generally of lesser importance. 242 4. Graph Theory Graph Theory Definitions. There are a lot of deﬁnitions to keep track of in graph theory. Here is a glossary of the terms we have already used and will soon encounter. Graph A collection of vertices, some of which are connected by edges. More precisely, a pair of sets V and E where V is a set of vertices and E is a set of 2-element subsets of V. Adjacent Two vertices are adjacent if they are connected by an edge. Two edges are adjacent if they share a vertex. Bipartite graph A graph for which it is possible to divide the vertices into two disjoint sets such that there are no edges between any two vertices in the same set. Complete bipartite graph A bipartite graph for which every vertex in the ﬁrst set is adjacent to every vertex in the second set. Complete graph A graph in which every pair of vertices is adjacent. Connected A graph is connected if there is a path from any vertex to any other vertex. Chromatic number The minimum number of colors required in a proper vertex coloring of the graph. Cycle A path (see below) that starts and stops at the same vertex, but contains no other repeated vertices. Degree of a vertex The number of edges incident to a vertex. Euler path A walk which uses each edge exactly once. Euler circuit An Euler path which starts and stops at the same vertex. Multigraph A multigraph is just like a graph but can contain multiple edges between two vertices as well as single edge loops (that is an edge from a vertex to itself). 4.1. Definitions 243 Path A path is a walk that doesn’t repeat any vertices (or edges) except perhaps the ﬁrst and last. If a path starts and ends at the same vertex, it is called a cycle. Planar A graph which can be drawn (in the plane) without any edges crossing. Subgraph We say that H is a subgraph of G if every vertex and edge of H is also a vertex or edge of G. We say H is an induced subgraph of G if every vertex of H is a vertex of G and each pair of vertices in H are adjacent in H if and only if they are adjacent in G. Tree A connected graph with no cycles. (If we remove the requirement that the graph is connected, the graph is called a forest.) The vertices in a tree with degree 1 are called leaves. Vertex coloring An assignment of colors to each of the vertices of a graph. A vertex coloring is proper if adjacent vertices are always colored diﬀerently. Walk A sequence of vertices such that consecutive vertices (in the sequence) are adjacent (in the graph). A walk in which no edge is repeated is called a trail, and a trail in which no vertex is repeated (except possibly the ﬁrst and last) is called a path. Exercises 1. If 10 people each shake hands with each other, how many handshakes took place? What does this question have to do with graph theory? 2. Among a group of 5 people, is it possible for everyone to be friends with exactly 2 of the people in the group? What about 3 of the people in the group? 3. Is it possible for two diﬀerent (non-isomorphic) graphs to have the same number of vertices and the same number of edges? What if the degrees of the vertices in the two graphs are the same (so both graphs have vertices with degrees 1, 2, 2, 3, and 4, for example)? Draw two such graphs or explain why not. 4. Are the two graphs below equal? Are they isomorphic? If they are isomorphic, give the isomorphism. If not, explain. Graph 1: V {a, b, c, d, e}, E {{a, b}, {a, c}, {a, e}, {b, d}, {b, e}, {c, d}} Graph 2: 244 4. Graph Theory a e b c d 5. Consider the following two graphs: G1 V1 {a, b, c, d, e , f , g} E1 {{a, b}, {a, d}, {b, c}, {b, d}, {b, e}, {b, f }, {c, g}, {d, e}, {e , f }, { f , g}}. G2 V2 {v1 , v 2 , v 3 , v 4 , v 5 , v 6 , v7 }, E2 {{v 1 , v4 }, {v 1 , v 5 }, {v 1 , v 7 }, {v2 , v 3 }, {v 2 , v 6 }, {v 3 , v 5 }, {v 3 , v 7 }, {v 4 , v 5 }, {v 5 , v6 }, {v 5 , v 7 }} (a) Let f : G1 → G2 be a function that takes the vertices of Graph 1 to vertices of Graph 2. The function is given by the following table: x f (x) a v4 b v5 c v1 d v6 e v2 f v3 g v7 Does f deﬁne an isomorphism between Graph 1 and Graph 2? (b) Deﬁne a new function g (with g , f ) that deﬁnes an isomorphism between Graph 1 and Graph 2. (c) Is the graph pictured below isomorphic to Graph 1 and Graph 2? Explain. 6. What is the largest number of edges possible in a graph with 10 vertices? What is the largest number of edges possible in a bipartite graph with 10 vertices? What is the largest number of edges possible in a tree with 10 vertices? 7. Which of the graphs below are bipartite? Justify your answers. 8. For which n ≥ 3 is the graph C n bipartite? 4.1. Definitions 9. 245 For each of the following, try to give two diﬀerent unlabeled graphs with the given properties, or explain why doing so is impossible. (a) Two diﬀerent trees with the same number of vertices and the same number of edges. A tree is a connected graph with no cycles. (b) Two diﬀerent graphs with 8 vertices all of degree 2. (c) Two diﬀerent graphs with 5 vertices all of degree 4. (d) Two diﬀerent graphs with 5 vertices all of degree 3. 10. Decide whether the statements below about subgraphs are true or false. For those that are true, brieﬂy explain why (1 or 2 sentences). For any that are false, give a counterexample. (a) Any subgraph of a complete graph is also complete. (b) Any induced subgraph of a complete graph is also complete. (c) Any subgraph of a bipartite graph is bipartite. (d) Any subgraph of a tree is a tree. 11. Let k 1 , k 2 , . . . , k j be a list of positive integers that sum to n (i.e, n). Use two graphs containing n vertices to explain why j Õ k i i1 2 Íj i1 ki ≤ n . 2 12. We often deﬁne graph theory concepts using set theory. For example, given a graph G (V, E) and a vertex v ∈ V, we deﬁne N(v) {u ∈ V : {v, u} ∈ E}. We deﬁne N[v] N(v) ∪ {v}. The goal of this problem is to ﬁgure out what all this means. (a) Let G be the graph with V {a, b, c, d, e , f } and E {{a, b}, {a, e}, {b, c}, {b, e}, {c, d}, {c, f }, {d, f }, {e, f }}. Find N(a), N[a], N(c), and N[c]. (b) What is the largest and smallest possible values for |N(v)| and |N[v]| for the graph in part (a)? Explain. (c) Give an example of a graph G (V, E) (probably diﬀerent than the one above) for which N[v] V for some vertex v ∈ V. Is there a graph for which N[v] V for all v ∈ V? Explain. (d) Give an example of a graph G (V, E) for which N(v) ∅ for some v ∈ V. Is there an example of such a graph for which N[u] V for some other u ∈ V as well? Explain. 246 4. Graph Theory (e) Describe in words what N(v) and N[v] mean in general. 13. A graph is a way of representing the relationships between elements in a set: an edge between the vertices x and y tells us that x is related to y (which we can write as x ∼ y). Not all sorts of relationships can be represented by a graph though. For each relationship described below, either draw the graph or explain why the relationship cannot be represented by a graph. (a) The set V {1, 2, . . . , 9} and the relationship x ∼ y when x − y is a non-zero multiple of 3. (b) The set V {1, 2, . . . , 9} and the relationship x ∼ y when y is a multiple of x. (c) The set V {1, 2, . . . , 9} and the relationship x ∼ y when 0 < |x − y| < 3. 14. Consider graphs with n vertices. Remember, graphs do not need to be connected. (a) How many edges must the graph have to guarantee at least one vertex has degree two or more? Prove your answer. (b) How many edges must the graph have to guarantee all vertices have degree two or more? Prove your answer. 15. Prove that any graph with at least two vertices must have two vertices of the same degree. 16. Suppose G is a connected graph with n > 1 vertices and n − 1 edges. Prove that G has a vertex of degree 1. 4.2. Trees 4.2 247 Trees Investigate! Consider the graph drawn below. 1. Find a subgraph with the smallest number of edges that is still connected and contains all the vertices. 2. Find a subgraph with the largest number of edges that doesn’t contain any cycles. 3. What do you notice about the number of edges in your examples above? Is this a coincidence? ! Attempt the above activity before proceeding ! One very useful and common approach to studying graph theory is to restrict your focus to graphs of a particular kind. For example, you could try to really understand just complete graphs or just bipartite graphs, instead of trying to understand all graphs in general. That is what we are going to do now, looking at trees. Hopefully by the end of this section we will have a better understanding of this class of graph, and also understand why it is important enough to warrant its own section. Deﬁnition of a Tree. A tree is a connected graph containing no cycles.4 A forest is a graph containing no cycles. Note that this means that a connected forest is a tree. Does the deﬁnition above agree with your intuition for what graphs we should call trees? Try thinking of examples of trees and make sure they satisfy the deﬁnition. One thing to keep in mind is that while the trees we study in graph theory are related to trees you might see in other subjects, the correspondence is not exact. For instance, in anthropology, you might study family trees, like the one below, 4Sometimes this is stated as “a tree is an acyclic connected graph;” “acyclic” is just a fancy word for “containing no cycles.” 248 4. Graph Theory Me Mom Dad Maternal Maternal Paternal Paternal Grandma Grandpa Grandma Grandpa So far so good, but while your grandparents are (probably) not bloodrelatives, if we go back far enough, it is likely that they did have some common ancestor. If you trace the tree back from you to that common ancestor, then down through your other grandparent, you would have a cycle, and thus the graph would not be a tree. You might also have seen something called a decision tree (such as the algorithm for deciding whether a series converges or diverges). Sometimes these too contain cycles, as the decision for one node might lead you back to a previous step. Both the examples of trees above also have another feature worth mentioning: there is a clear order to the vertices in the tree. In general, there is no reason for a tree to have this added structure, although we can impose such a structure by considering rooted trees, where we simply designate one vertex as the root. We will consider such trees in more detail later in this section. Properties of Trees We wish to really understand trees. This means we should discover properties of trees; what makes them special and what is special about them. A tree is a connected graph with no cycles. Is there anything else we can say? It would be nice to have other equivalent conditions for a graph to be a tree. That is, we would like to know whether there are any graph theoretic properties that all trees have, and perhaps even that only trees have. To get a feel for the sorts of things we can say, we will consider three propositions about trees. These will also illustrate important proof techniques that apply to graphs in general, and happen to be a little easier for trees. Our ﬁrst proposition gives an alternate deﬁnition for a tree. That is, it gives necessary and suﬃcient conditions for a graph to be a tree. Proposition 4.2.1 A graph T is a tree if and only if between every pair of distinct vertices of T there is a unique path. 4.2. Trees 249 Proof. This is an “if and only if” statement, so we must prove two implications. We start by proving that if T is a tree, then between every pair of distinct vertices there is a unique path. Assume T is a tree, and let u and v be distinct vertices (if T only has one vertex, then the conclusion is satisﬁed automatically). We must show two things to show that there is a unique path between u and v: that there is a path, and that there is not more than one path. The ﬁrst of these is automatic, since T is a tree, it is connected, so there is a path between any pair of vertices. To show the path is unique, we suppose there are two paths between u and v, and get a contradiction. The two paths might start out the same, but since they are diﬀerent, there is some ﬁrst vertex u ′ after which the two paths diverge. However, since the two paths both end and v, there is some ﬁrst vertex after u ′ that they have in common, call it v ′. Now consider the two paths from u ′ to v ′. Taken together, these form a cycle, which contradicts our assumption that T is a tree. Now we consider the converse: if between every pair of distinct vertices of T there is a unique path, then T is a tree. So assume the hypothesis: between every pair of distinct vertices of T there is a unique path. To prove that T is a tree, we must show it is connected and contains no cycles. The ﬁrst half of this is easy: T is connected, because there is a path between every pair of vertices. To show that T has no cycles, we assume it does, for the sake of contradiction. Let u and v be two distinct vertices in a cycle of T. Since we can get from u to v by going clockwise or counterclockwise around the cycle, there are two paths from u and v, contradicting our assumption. We have established both directions so we have completed the proof. qed Read the proof above very carefully. Notice that both directions had two parts: the existence of paths, and the uniqueness of paths (which related to the fact that there were no cycles). In this case, these two parts were really separate. In fact, if we just considered graphs with no cycles (a forest), then we could still do the parts of the proof that explore the uniqueness of paths between vertices, even if there might not exist paths between vertices. This observation allows us to state the following corollary:5 Corollary 4.2.2 A graph F is a forest if and only if between any pair of vertices in F there is at most one path. We do not give a proof of the corollary (it is, after all, supposed to follow directly from the proposition) but for practice, you are asked to 5A corollary is another sort of provable statement, like a proposition or theorem, but one that follows direction from another already established statement, or its proof. 250 4. Graph Theory give a careful proof in the exercises. When you do so, try to use proof by contrapositive instead of proof by contradiction. Our second proposition tells us that all trees have leaves: vertices of degree one. Proposition 4.2.3 Any tree with at least two vertices has at least two vertices of degree one. Proof. We give a proof by contradiction. Let T be a tree with at least two vertices, and suppose, contrary to stipulation, that there are not two vertices of degree one. Let P be a path in T of longest possible length. Let u and v be the endpoints of the path. Since T does not have two vertices of degree one, at least one of these must have degree two or higher. Say that it is u. We know that u is adjacent to a vertex in the path P, but now it must also be adjacent to another vertex, call it u ′. Where is u ′? It cannot be a vertex of P, because if it was, there would be two distinct paths from u to u ′: the edge between them, and the ﬁrst part of P (up to u ′). But u ′ also cannot be outside of P, for if it was, there would be a path from u ′ to v that was longer than P, which has longest possible length. This contradiction proves that there must be at least two vertices of degree one. In fact, we can say a little more: u and v must both have degree one. qed The proposition is quite useful when proving statements about trees, because we often prove statements about trees by induction. To do so, we need to reduce a given tree to a smaller tree (so we can apply the inductive hypothesis). Getting rid of a vertex of degree one is an obvious choice, and now we know there is always one to get rid of. To illustrate how induction is used on trees, we will consider the relationship between the number of vertices and number of edges in trees. Is there a tree with exactly 7 vertices and 7 edges? Try to draw one. Could a tree with 7 vertices have only 5 edges? There is a good reason that these seem impossible to draw. Proposition 4.2.4 Let T be a tree with v vertices and e edges. Then e v − 1. Proof. We will give a proof by induction on the number of vertices in the tree. That is, we will prove that every tree with v vertices has exactly v − 1 edges, and then use induction to show this is true for all v ≥ 1. For the base case, consider all trees with v 1 vertices. There is only one such tree: the graph with a single isolated vertex. This graph has e 0 edges, so we see that e v − 1 as needed. Now for the inductive case, ﬁx k ≥ 1 and assume that all trees with v k vertices have exactly e k − 1 edges. Now consider an arbitrary tree T with v k + 1 vertices. By Proposition 4.2.3, T has a vertex v 0 of degree 4.2. Trees 251 one. Let T ′ be the tree resulting from removing v 0 from T (together with its incident edge). Since we removed a leaf, T ′ is still a tree (the unique paths between pairs of vertices in T ′ are the same as the unique paths between them in T). Now T ′ has k vertices, so by the inductive hypothesis, has k − 1 edges. What can we say about T? Well, it has one more edge than T ′, so it has k edges. But this is exactly what we wanted: v k + 1, e k so indeed e v − 1. This completes the inductive case, and the proof. qed There is a very important feature of this induction proof that is worth noting. Induction makes sense for proofs about graphs because we can think of graphs as growing into larger graphs. However, this does NOT work. It would not be correct to start with a tree with k vertices, and then add a new vertex and edge to get a tree with k + 1 vertices, and note that the number of edges also grew by one. Why is this bad? Because how do you know that every tree with k + 1 vertices is the result of adding a vertex to your arbitrary starting tree? You don’t! The point is that whenever you give an induction proof that a statement about graphs that holds for all graphs with v vertices, you must start with an arbitrary graph with v + 1 vertices, then reduce that graph to a graph with v vertices, to which you can apply your inductive hypothesis. Rooted Trees So far, we have thought of trees only as a particular kind of graph. However, it is often useful to add additional structure to trees to help solve problems. Data is often structured like a tree. This book, for example, has a tree structure: draw a vertex for the book itself. Then draw vertices for each chapter, connected to the book vertex. Under each chapter, draw a vertex for each section, connecting it to the chapter it belongs to. The graph will not have any cycles; it will be a tree. But a tree with clear hierarchy which is not present if we don’t identify the book vertex as the “top”. As soon as one vertex of a tree is designated as the root, then every other vertex on the tree can be characterized by its position relative to the root. This works because there is a unique path between any two vertices in a tree. So from any vertex, we can travel back to the root in exactly one way. This also allows us to describe how distinct vertices in a rooted tree are related. If two vertices are adjacent, then we say one of them is the parent of the other, which is called the child of the parent. Of the two, the parent is the vertex that is closer to the root. Thus the root of a tree is a parent, but is not the child of any vertex (and is unique in this respect: all non-root vertices have exactly one parent). 252 4. Graph Theory Not surprisingly, the child of a child of a vertex is called the grandchild of the vertex (and it is the grandparent). More in general, we say that a vertex v is a descendent of a vertex u provided u is a vertex on the path from v to the root. Then we would call u an ancestor of v. For most trees (in fact, all except paths with one end the root), there will be pairs of vertices neither of which is a descendant of the other. We might call these cousins or siblings. In fact, vertices u and v are called siblings provided they have the same parent. Note that siblings are never adjacent (do you see why?). Example 4.2.5 Consider the tree below. b a c d f e h i g If we designate vertex f as the root, then e, h, and i are the children of f , and are siblings of each other. Among the other things we cay say are that a is a child of c, and a descendant of f . The vertex g is a descendant of f , in fact, is a grandchild of f . Vertices g and d are siblings, since they have the common parent e. Notice how this changes if we pick a diﬀerent vertex for the root. If a is the root, then its lone child is c, which also has only one child, namely e. We would then have f the child of e (instead of the other way around), and f is the descendant of a, instead of the ancestor. f and g are now siblings. All of this ﬂowery language helps us describe how to navigate through a tree. Traversing a tree, visiting each vertex in some order, is a key step in many algorithms. Even if the tree is not rooted, we can always form a rooted tree by picking any vertex as the root. Here is an example of why doing so can be helpful. Example 4.2.6 Explain why every tree is a bipartite graph. Solution. To show that a graph is bipartite, we must divide the vertices into two sets A and B so that no two vertices in the same set are adjacent. Here is an algorithm that does just this. Designate any vertex as the root. Put this vertex in set A. Now put all of the children of the root in set B. None of these children 4.2. Trees 253 are adjacent (they are siblings), so we are good so far. Now put into A every child of every vertex in B (i.e., every grandchild of the root). Keep going until all vertices have been assigned one of the sets, alternating between A and B every “generation.” That is, a vertex is in set B if and only if it is the child of a vertex in set A. The key to how we partitioned the tree in the example was to know which vertex to assign to a set next. We chose to visit all vertices in the same generation before any vertices of the next generation. This is usually called a breadth ﬁrst search (we say “search” because you often traverse a tree looking for vertices with certain properties). In contrast, we could also have partitioned the tree in a diﬀerent order. Start with the root, put it in A. Then look for one child of the root to put in B. Then ﬁnd a child of that vertex, into A, and then ﬁnd its child, into B, and so on. When you get to a vertex with no children, retreat to its parent and see if the parent has any other children. So we travel as far from the root as fast as possible, then backtrack until we can move forward again. This is called depth ﬁrst search. These algorithmic explanations can serve as a proof that every tree is bipartite, although care needs to be spent to prove that the algorithms are correct. Another approach to prove that all trees are bipartite, using induction, is requested in the exercises. Spanning Trees One of the advantages of trees is that they give us a few simple ways to travel through the vertices. If a connected graph is not a tree, then we can still use these traversal algorithms if we identify a subgraph that is a tree. First we should consider if this even makes sense. Given any connected graph G, will there always be a subgraph that is a tree? Well, that is actually too easy: you could just take a single edge of G. If we want to use this subgraph to tell us how to visit all vertices, then we want our subgraph to include all of the vertices. We call such a tree a spanning tree. It turns out that every connected graph has one (and usually many). Spanning tree. Given a connected graph G, a spanning tree of G is a subgraph of G which is a tree and includes all the vertices of G. Every connected graph has a spanning tree. How do we know? We can give an algorithm for ﬁnding a spanning tree! Start with a connected graph G. If there is no cycle, then G is already a tree and we are done. If there is a cycle, let e be any edge in that cycle 254 4. Graph Theory and consider the new graph G1 G − e (i.e., the graph you get by deleting e). This tree is still connected since e belonged to a cycle, there were at least two paths between its incident vertices. Now repeat: if G1 has no cycles, we are done, otherwise deﬁne G2 to be G1 − e1 , where e1 is an edge in a cycle in G1 . Keep going. This process must eventually stop, since there are only a ﬁnite number of edges to remove. The result will be a tree, and since we never removed any vertex, a spanning tree. This is by no means the only algorithm for ﬁnding a spanning tree. You could have started with the empty graph and added edges that belong to G as long as adding them would not create a cycle. You have some choices as to which edges you add ﬁrst: you could always add an edge adjacent to edges you have already added (after the ﬁrst one, of course), or add them using some other order. Which spanning tree you end up with depends on these choices. Example 4.2.7 Find two diﬀerent spanning trees of the graph, Solution. Here are two spanning trees. Although we will not consider this in detail, these algorithms are usually applied to weighted graphs. Here every edge has some weight or cost assigned to it. The goal is to ﬁnd a spanning tree that has the smallest possible combined weight. Such a tree is called a minimum spanning tree. Finding the minimum spanning tree uses basically the same algorithms as we described above, but when picking an edge to add, you always pick the smallest (or when removing an edge, you always remove the largest).6 6If you add the smallest edge adjacent to edges you have already added, you are doing Prim’s algorithm. If you add the smallest edge in the entire graph, you are following Kruskal’s algorithm. 4.2. Trees 255 Exercises 1. Which of the following graphs are trees? (a) G (V, E) with V {a, b, c, d, e} and E {{a, b}, {a, e}, {b, c}, {c, d}, {d, e}} (b) G (V, E) with V {a, b, c, d, e} and E {{a, b}, {b, c}, {c, d}, {d, e}} (c) G (V, E) with V {a, b, c, d, e} and E {{a, b}, {a, c}, {a, d}, {a, e}} (d) G (V, E) with V {a, b, c, d, e} and E {{a, b}, {a, c}, {d, e}} 2. For each degree sequence below, decide whether it must always, must never, or could possibly be a degree sequence for a tree. Remember, a degree sequence lists out the degrees (number of edges incident to the vertex) of all the vertices in a graph in non-increasing order. (a) (4, 1, 1, 1, 1) (b) (3, 3, 2, 1, 1) (c) (2, 2, 2, 1, 1) (d) (4, 4, 3, 3, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1) 3. For each degree sequence below, decide whether it must always, must never, or could possibly be a degree sequence for a tree. Justify your answers. (a) (3, 3, 2, 2, 2) (b) (3, 2, 2, 1, 1, 1) (c) (3, 3, 3, 1, 1, 1) (d) (4, 4, 1, 1, 1, 1, 1, 1) 4. Suppose you have a graph with v vertices and e edges that satisﬁes v e + 1. Must the graph be a tree? Prove your answer. 5. Prove that any graph (not necessarily a tree) with v vertices and e edges that satisﬁes v > e + 1 will NOT be connected. 6. If a graph G with v vertices and e edges is connected and has v < e +1, must it contain a cycle? Prove your answer. 7. We deﬁne a forest to be a graph with no cycles. (a) Explain why this is a good name. That is, explain why a forest is a union of trees. 256 4. Graph Theory (b) Suppose F is a forest consisting of m trees and v vertices. How many edges does F have? Explain. (c) Prove that any graph G with v vertices and e edges that satisﬁes v < e + 1 must contain a cycle (i.e., not be a forest). 8. Give a careful proof of Corollary 4.2.2: A graph is a forest if and only if there is at most one path between any pair of vertices. Use proof by contrapositive (and not a proof by contradiction) for both directions. 9. Give a careful proof by induction on the number of vertices, that every tree is bipartite. 10. Consider the tree drawn below. c a g d b e h f i (a) Suppose we designate vertex e as the root. List the children, parents and siblings of each vertex. Does any vertex other than e have grandchildren? (b) Suppose e is not chosen as the root. Does our choice of root vertex change the number of children e has? The number of grandchildren? How many are there of each? (c) In fact, pick any vertex in the tree and suppose it is not the root. Explain why the number of children of that vertex does not depend on which other vertex is the root. (d) Does the previous part work for other trees? Give an example of a diﬀerent tree for which it holds. Then either prove that it always holds or give an example of a tree for which it doesn’t. 11. Let T be a rooted tree that contains vertices u, v, and w (among possibly others). Prove that if w is a descendant of both u and v, then u is a descendant of v or v is a descendant of u. 12. Unless it is already a tree, a given graph G will have multiple spanning trees. How similar or diﬀerent must these be? (a) Must all spanning trees of a given graph be isomorphic to each other? Explain why or give a counterexample. (b) Must all spanning trees of a given graph have the same number of edges? Explain why or give a counterexample. 4.2. Trees 257 (c) Must all spanning trees of a graph have the same number of leaves (vertices of degree 1)? Explain why or give a counterexample. 13. Find all spanning trees of the graph below. How many diﬀerent spanning trees are there? How many diﬀerent spanning trees are there up to isomorphism (that is, if you grouped all the spanning trees by which are isomorphic, how many groups would you have)? a d e c b f 14. Give an example of a graph that has exactly 7 diﬀerent spanning trees. Note, it acceptable for some or all of these spanning trees to be isomorphic. 15. Prove that every connected graph which is not itself a tree must have at last three diﬀerent (although possibly isomorphic) spanning trees. 16. Consider edges that must be in every spanning tree of a graph. Must every graph have such an edge? Give an example of a graph that has exactly one such edge. 258 4. Graph Theory 4.3 Planar Graphs Investigate! When a connected graph can be drawn without any edges crossing, it is called planar. When a planar graph is drawn in this way, it divides the plane into regions called faces. 1. Draw, if possible, two diﬀerent planar graphs with the same number of vertices, edges, and faces. 2. Draw, if possible, two diﬀerent planar graphs with the same number of vertices and edges, but a diﬀerent number of faces. ! Attempt the above activity before proceeding ! When is it possible to draw a graph so that none of the edges cross? If this is possible, we say the graph is planar (since you can draw it on the plane). Notice that the deﬁnition of planar includes the phrase “it is possible to.” This means that even if a graph does not look like it is planar, it still might be. Perhaps you can redraw it in a way in which no edges cross. For example, this is a planar graph: That is because we can redraw it like this: The graphs are the same, so if one is planar, the other must be too. However, the original drawing of the graph was not a planar representation of the graph. When a planar graph is drawn without edges crossing, the edges and vertices of the graph divide the plane into regions. We will call each region a face. The graph above has 3 faces (yes, we do include the “outside” region as a face). The number of faces does not change no matter how you draw the graph (as long as you do so without the edges 4.3. Planar Graphs 259 crossing), so it makes sense to ascribe the number of faces as a property of the planar graph. WARNING: you can only count faces when the graph is drawn in a planar way. For example, consider these two representations of the same graph: If you try to count faces using the graph on the left, you might say there are 5 faces (including the outside). But drawing the graph with a planar representation shows that in fact there are only 4 faces. There is a connection between the number of vertices (v), the number of edges (e) and the number of faces ( f ) in any connected planar graph. This relationship is called Euler’s formula. Euler’s Formula for Planar Graphs. For any (connected) planar graph with v vertices, e edges and f faces, we have v − e + f 2. Why is Euler’s formula true? One way to convince yourself of its validity is to draw a planar graph step by step. Start with the graph P2 : Any connected graph (besides just a single isolated vertex) must contain this subgraph. Now build up to your graph by adding edges and vertices. Each step will consist of either adding a new vertex connected by a new edge to part of your graph (so creating a new “spike”) or by connecting two vertices already in the graph with a new edge (completing a circuit). What do these “moves” do? When adding the spike, the number of edges increases by 1, the number of vertices increases by one, and the number of faces remains the same. But this means that v − e + f does not 260 4. Graph Theory change. Completing a circuit adds one edge, adds one face, and keeps the number of vertices the same. So again, v − e + f does not change. Since we can build any graph using a combination of these two moves, and doing so never changes the quantity v − e + f , that quantity will be the same for all graphs. But notice that our starting graph P2 has v 2, e 1 and f 1, so v − e + f 2. This argument is essentially a proof by induction. A good exercise would be to rewrite it as a formal induction proof. Non-planar Graphs Investigate! For the complete graphs K n , we would like to be able to say something about the number of vertices, edges, and (if the graph is planar) faces. Let’s ﬁrst consider K 3 : 1. How many vertices does K 3 have? How many edges? 2. If K3 is planar, how many faces should it have? Repeat parts (1) and (2) for K 4 , K5 , and K23 . What about complete bipartite graphs? How many vertices, edges, and faces (if it were planar) does K7,4 have? For which values of m and n are K n and K m,n planar? ! Attempt the above activity before proceeding ! Not all graphs are planar. If there are too many edges and too few vertices, then some of the edges will need to intersect. The smallest graph where this happens is K5 . If you try to redraw this without edges crossing, you quickly get into trouble. There seems to be one edge too many. In fact, we can prove that no matter how you draw it, K 5 will always have edges crossing. Theorem 4.3.1 K5 is not planar. Proof. The proof is by contradiction. So assume that K 5 is planar. Then the graph must satisfy Euler’s formula for planar graphs. K 5 has 5 vertices and 10 edges, so we get 5 − 10 + f 2, 4.3. Planar Graphs 261 which says that if the graph is drawn without any edges crossing, there would be f 7 faces. Now consider how many edges surround each face. Each face must be surrounded by at least 3 edges. Let B be the total number of boundaries around all the faces in the graph. Thus we have that 3 f ≤ B. But also B 2e, since each edge is used as a boundary exactly twice. Putting this together we get 3 f ≤ 2e. But this is impossible, since we have already determined that f 7 and e 10, and 21 ̸ ≤ 20. This is a contradiction so in fact K 5 is not planar. qed The other simplest graph which is not planar is K3,3 Proving that K3,3 is not planar answers the houses and utilities puzzle: it is not possible to connect each of three houses to each of three utilities without the lines crossing. Theorem 4.3.2 K 3,3 is not planar. Proof. Again, we proceed by contradiction. Suppose K3,3 were planar. Then by Euler’s formula there will be 5 faces, since v 6, e 9, and 6 − 9 + f 2. How many boundaries surround these 5 faces? Let B be this number. Since each edge is used as a boundary twice, we have B 2e. Also, B ≥ 4 f since each face is surrounded by 4 or more boundaries. We know this is true because K 3,3 is bipartite, so does not contain any 3-edge cycles. Thus 4 f ≤ 2e. But this would say that 20 ≤ 18, which is clearly false. Thus K3,3 is not planar. qed Note the similarities and diﬀerences in these proofs. Both are proofs by contradiction, and both start with using Euler’s formula to derive the (supposed) number of faces in the graph. Then we ﬁnd a relationship between the number of faces and the number of edges based on how many edges surround each face. This is the only diﬀerence. In the proof for K 5 , we got 3 f ≤ 2e and for K 3,3 we go 4 f ≤ 2e. The coeﬃcient of f is the key. It is the smallest number of edges which could surround any face. If some number of edges surround a face, then these edges form a cycle. So that number is the size of the smallest cycle in the graph. In general, if we let g be the size of the smallest cycle in a graph (g stands for girth, which is the technical term for this) then for any planar 262 4. Graph Theory graph we have g f ≤ 2e. When this disagrees with Euler’s formula, we know for sure that the graph cannot be planar. Polyhedra Investigate! A cube is an example of a convex polyhedron. It contains 6 identical squares for its faces, 8 vertices, and 12 edges. The cube is a regular polyhedron (also known as a Platonic solid) because each face is an identical regular polygon and each vertex joins an equal number of faces. There are exactly four other regular polyhedra: the tetrahedron, octahedron, dodecahedron, and icosahedron with 4, 8, 12 and 20 faces respectively. How many vertices and edges do each of these have? ! Attempt the above activity before proceeding ! Another area of mathematics where you might have heard the terms “vertex,” “edge,” and “face” is geometry. A polyhedron is a geometric solid made up of ﬂat polygonal faces joined at edges and vertices. We are especially interested in convex polyhedra, which means that any line segment connecting two points on the interior of the polyhedron must be entirely contained inside the polyhedron.7 Notice that since 8 − 12 + 6 2, the vertices, edges and faces of a cube satisfy Euler’s formula for planar graphs. This is not a coincidence. We can represent a cube as a planar graph by projecting the vertices and edges onto the plane. One such projection looks like this: In fact, every convex polyhedron can be projected onto the plane without edges crossing. Think of placing the polyhedron inside a sphere, with a light at the center of the sphere. The edges and vertices of the polyhedron cast a shadow onto the interior of the sphere. You can then cut a hole in the sphere in the middle of one of the projected faces and “stretch” the 7An alternative deﬁnition for convex is that the internal angle formed by any two faces must be less than 180 deg. 4.3. Planar Graphs 263 sphere to lie down ﬂat on the plane. The face that was punctured becomes the “outside” face of the planar graph. The point is, we can apply what we know about graphs (in particular planar graphs) to convex polyhedra. Since every convex polyhedron can be represented as a planar graph, we see that Euler’s formula for planar graphs holds for all convex polyhedra as well. We also can apply the same sort of reasoning we use for graphs in other contexts to convex polyhedra. For example, we know that there is no convex polyhedron with 11 vertices all of degree 3, as this would make 33/2 edges. Example 4.3.3 Is there a convex polyhedron consisting of three triangles and six pentagons? What about three triangles, six pentagons and ﬁve heptagons (7-sided polygons)? Solution. How many edges would such polyhedra have? For the ﬁrst proposed polyhedron, the triangles would contribute a total of 9 edges, and the pentagons would contribute 30. However, this counts each edge twice (as each edge borders exactly two faces), giving 39/2 edges, an impossibility. There is no such polyhedron. The second polyhedron does not have this obstacle. The extra 35 edges contributed by the heptagons give a total of 74/2 = 37 edges. So far so good. Now how many vertices does this supposed polyhedron have? We can use Euler’s formula. There are 14 faces, so we have v − 37 + 14 2 or equivalently v 25. But now use the vertices to count the edges again. Each vertex must have degree at least three (that is, each vertex joins at least three faces since the interior angle of all the polygons must be less that 180◦ ), so the sum of the degrees of vertices is at least 75. Since the sum of the degrees must be exactly twice the number of edges, this says that there are strictly more than 37 edges. Again, there is no such polyhedron. To conclude this application of planar graphs, consider the regular polyhedra. We claimed there are only ﬁve. How do we know this is true? We can prove it using graph theory. Theorem 4.3.4 There are exactly ﬁve regular polyhedra. Proof. Recall that all the faces of a regular polyhedron are identical regular polygons, and that each vertex has the same degree. Consider four cases, depending on the type of regular polygon. Case 1: Each face is a triangle. Let f be the number of faces. There are then 3 f /2 edges. Using Euler’s formula we have v − 3 f /2 + f 2 so v 2 + f /2. Now each vertex has the same degree, say k. So the number 264 4. Graph Theory of edges is also kv/2. Putting this together gives 3f k(2 + f /2) , 2 2 e which says k 6f . 4+ f 6f Both k and f must be positive integers. Note that 4+ f is an increasing function for positive f , bounded above by a horizontal asymptote at f 6. Thus the only possible values for k are 3, 4, and 5. Each of these are possible. To get k 3, we need f 4 (this is the tetrahedron). For k 4 we take f 8 (the octahedron). For k 5 take f 20 (the icosahedron). Thus there are exactly three regular polyhedra with triangles for faces. Case 2: Each face is a square. Now we have e 4 f /2 2 f . Using Euler’s formula we get v 2 + f , and counting edges using the degree k of each vertex gives us k(2 + f ) e 2f . 2 Solving for k gives k 4f 8f . 2+ f 4+2f This is again an increasing function, but this time the horizontal asymptote is at k 4, so the only possible value that k could take is 3. This produces 6 faces, and we have a cube. There is only one regular polyhedron with square faces. Case 3: Each face is a pentagon. We perform the same calculation as above, this time getting e 5 f /2 so v 2 + 3 f /2. Then e 5f k(2 + 3 f /2) , 2 2 so k 10 f . 4+3f Now the horizontal asymptote is at 10 3 . This is less than 4, so we can only hope of making k 3. We can do so by using 12 pentagons, getting the dodecahedron. This is the only regular polyhedron with pentagons as faces. Case 4: Each face is an n-gon with n ≥ 6. Following the same procedure as above, we deduce that k 2n f , 4 + (n − 2) f 4.3. Planar Graphs 265 2n which will be increasing to a horizontal asymptote of n−2 . When n 6, this asymptote is at k 3. Any larger value of n will give an even smaller asymptote. Therefore no regular polyhedra exist with faces larger than pentagons.a qed a Notice that you can tile the plane with hexagons. This is an inﬁnite planar graph; each vertex has degree 3. These inﬁnitely many hexagons correspond to the limit as f → ∞ to make k 3. Exercises 1. Is it possible for a planar graph to have 6 vertices, 10 edges and 5 faces? Explain. 2. The graph G has 6 vertices with degrees 2, 2, 3, 4, 4, 5. How many edges does G have? Could G be planar? If so, how many faces would it have. If not, explain. 3. Is it possible for a connected graph with 7 vertices and 10 edges to be drawn so that no edges cross and create 4 faces? Explain. 4. Is it possible for a graph with 10 vertices and edges to be a connected planar graph? Explain. 5. Is there a connected planar graph with an odd number of faces where every vertex has degree 6? Prove your answer. 6. I’m thinking of a polyhedron containing 12 faces. Seven are triangles and four are quadralaterals. The polyhedron has 11 vertices including those around the mystery face. How many sides does the last face have? 7. Consider some classic polyhedrons. (a) An octahedron is a regular polyhedron made up of 8 equilateral triangles (it sort of looks like two pyramids with their bases glued together). Draw a planar graph representation of an octahedron. How many vertices, edges and faces does an octahedron (and your graph) have? (b) The traditional design of a soccer ball is in fact a (spherical projection of a) truncated icosahedron. This consists of 12 regular pentagons and 20 regular hexagons. No two pentagons are adjacent (so the edges of each pentagon are shared only by hexagons). How many vertices, edges, and faces does a truncated icosahedron have? Explain how you arrived at your answers. Bonus: draw the planar graph representation of the truncated icosahedron. 266 4. Graph Theory (c) Your “friend” claims that he has constructed a convex polyhedron out of 2 triangles, 2 squares, 6 pentagons and 5 octagons. Prove that your friend is lying. Hint: each vertex of a convex polyhedron must border at least three faces. 8. Prove Euler’s formula using induction on the number of edges in the graph. 9. Prove Euler’s formula using induction on the number of vertices in the graph. 10. Euler’s formula (v − e + f 2) holds for all connected planar graphs. What if a graph is not connected? Suppose a planar graph has two components. What is the value of v − e + f now? What if it has k components? 11. Prove that the Petersen graph (below) is not planar. 12. Prove that any planar graph with v vertices and e edges satisﬁes e ≤ 3v − 6. 13. Prove that any planar graph must have a vertex of degree 5 or less. 14. Give a careful proof that the graph below is not planar. 15. Explain why we cannot use the same sort of proof we did in Exercise 4.3.14 to prove that the graph below is not planar. Then explain how you know the graph is not planar anyway. 4.4. Coloring 4.4 267 Coloring Investigate! Mapmakers in the ﬁctional land of Euleria have drawn the borders of the various dukedoms of the land. To make the map pretty, they wish to color each region. Adjacent regions must be colored diﬀerently, but it is perfectly ﬁne to color two distant regions with the same color. What is the fewest colors the mapmakers can use and still accomplish this task? ! Attempt the above activity before proceeding ! Perhaps the most famous graph theory problem is how to color maps. Given any map of countries, states, counties, etc., how many colors are needed to color each region on the map so that neighboring regions are colored diﬀerently? Actual map makers usually use around seven colors. For one thing, they require watery regions to be a speciﬁc color, and with a lot of colors it is easier to ﬁnd a permissible coloring. We want to know whether there is a smaller palette that will work for any map. How is this related to graph theory? Well, if we place a vertex in the center of each region (say in the capital of each state) and then connect two vertices if their states share a border, we get a graph. Coloring regions on the map corresponds to coloring the vertices of the graph. Since neighboring regions cannot be colored the same, our graph cannot have vertices colored the same when those vertices are adjacent. 268 4. Graph Theory In general, given any graph G, a coloring of the vertices is called (not surprisingly) a vertex coloring. If the vertex coloring has the property that adjacent vertices are colored diﬀerently, then the coloring is called proper. Every graph has a proper vertex coloring. For example, you could color every vertex with a diﬀerent color. But often you can do better. The smallest number of colors needed to get a proper vertex coloring is called the chromatic number of the graph, written χ(G) . Example 4.4.1 Find the chromatic number of the graphs below. Solution. The graph on the left is K6 . The only way to properly color the graph is to give every vertex a diﬀerent color (since every vertex is adjacent to every other vertex). Thus the chromatic number is 6. The middle graph can be properly colored with just 3 colors (Red, Blue, and Green). For example: B G R R B G There is no way to color it with just two colors, since there are three vertices mutually adjacent (i.e., a triangle). Thus the chromatic number is 3. The graph on the right is just K 2,3 . As with all bipartite graphs, this graph has chromatic number 2: color the vertices on the top row red and the vertices on the bottom row blue. It appears that there is no limit to how large chromatic numbers can get. It should not come as a surprise that K n has chromatic number n. So how could there possibly be an answer to the original map coloring question? If the chromatic number of graph can be arbitrarily large, then it seems like there would be no upper bound to the number of colors needed for any map. But there is. The key observation is that while it is true that for any number n, there is a graph with chromatic number n, only some graphs arrive as representations of maps. If you convert a map to a graph, the edges 4.4. Coloring 269 between vertices correspond to borders between the countries. So you should be able to connect vertices in such a way where the edges do not cross. In other words, the graphs representing maps are all planar! So the question is, what is the largest chromatic number of any planar graph? The answer is the best known theorem of graph theory: Theorem 4.4.2 The Four Color Theorem. If G is a planar graph, then the chromatic number of G is less than or equal to 4. Thus any map can be properly colored with 4 or fewer colors. We will not prove this theorem. Really. Even though the theorem is easy to state and understand, the proof is not. In fact, there is currently no “easy” known proof of the theorem. The current best proof still requires powerful computers to check an unavoidable set of 633 reducible conﬁgurations. The idea is that every graph must contain one of these reducible conﬁgurations (this fact also needs to be checked by a computer) and that reducible conﬁgurations can, in fact, be colored in 4 or fewer colors. Coloring in General Investigate! The math department plans to oﬀer 10 classes next semester. Some classes cannot run at the same time (perhaps they are taught by the same professor, or are required for seniors). Class: Conﬂicts with: A DI DIJ B C EFI D ABF E CHI F CDI G J H EIJ ABCEFH I J BGH How many diﬀerent time slots are needed to teach these classes (and which should be taught at the same time)? More importantly, how could we use graph coloring to answer this question? ! Attempt the above activity before proceeding ! 270 4. Graph Theory Cartography is certainly not the only application of graph coloring. There are plenty of situations in which you might wish partition the objects in question so that related objects are not in the same set. For example, you might wish to store chemicals safely. To avoid explosions, certain pairs of chemicals should not be stored in the same room. By coloring a graph (with vertices representing chemicals and edges representing potential negative interactions), you can determine the smallest number of rooms needed to store the chemicals. Here is a further example: Example 4.4.3 Radio stations broadcast their signal at certain frequencies. However, there are a limited number of frequencies to choose from, so nationwide many stations use the same frequency. This works because the stations are far enough apart that their signals will not interfere; no one radio could pick them up at the same time. Suppose 10 new radio stations are to be set up in a currently unpopulated (by radio stations) region. The radio stations that are close enough to each other to cause interference are recorded in the table below. What is the fewest number of frequencies the stations could use. KQEA KQEB x x KQEA KQEB KQEC KQEC x x x x x KQEG x x x x x KQEH x x KQEJ x x x x x x x x x x x KQEI KQEI x x KQEH KQEJ KQEF x x KQEE KQEG KQEE x KQED KQEF KQED x x x x x Solution. Represent the problem as a graph with vertices as the stations and edges when two stations are close enough to cause interference. We are looking for the chromatic number of the graph. Vertices that are colored identically represent stations that can have the same frequency. This graph has chromatic number 5. A proper 5-coloring is shown on the right. Notice that the graph contains a copy of the complete graph K5 so no fewer than 5 colors can be used. 4.4. Coloring KQEA KQEJ P KQEB R G KQEI KQEC B B KQEH KQED R B Y KQEG KQEE KQEF G 271 G In the example above, the chromatic number was 5, but this is not a counterexample to the Four Color Theorem, since the graph representing the radio stations is not planar. It would be nice to have some quick way to ﬁnd the chromatic number of a (possibly non-planar) graph. It turns out nobody knows whether an eﬃcient algorithm for computing chromatic numbers exists. While we might not be able to ﬁnd the exact chromatic number of graph easily, we can often give a reasonable range for the chromatic number. In other words, we can give upper and lower bounds for chromatic number. This is actually not very diﬃcult: for every graph G, the chromatic number of G is at least 1 and at most the number of vertices of G. What? You want better bounds on the chromatic number? Well you are in luck. A clique in a graph is a set of vertices all of which are pairwise adjacent. In other words, a clique of size n is just a copy of the complete graph K n . We deﬁne the clique number of a graph to be the largest n for which the graph contains a clique of size n. Any clique of size n cannot be colored with fewer than n colors, so we have a nice lower bound: Theorem 4.4.4 The chromatic number of a graph G is at least the clique number of G. There are times when the chromatic number of G is equal to the clique number. These graphs have a special name; they are called perfect. If you know that a graph is perfect, then ﬁnding the chromatic number is simply a matter of searching for the largest clique.8 However, not all graphs are perfect. For an upper bound, we can improve on “the number of vertices” by looking to the degrees of vertices. Let ∆(G) be the largest degree of any vertex in the graph G. One reasonable guess for an upper bound on the chromatic number is χ(G) ≤ ∆(G) + 1. Why is this reasonable? Starting 8There are special classes of graphs which can be proved to be perfect. One such class is the set of chordal graphs, which have the property that every cycle in the graph contains a chord—an edge between two vertices in of the cycle which are not adjacent in the cycle. 272 4. Graph Theory with any vertex, it together with all of its neighbors can always be colored in ∆(G)+1 colors, since at most we are talking about ∆(G)+1 vertices in this set. Now fan out! At any point, if you consider an already colored vertex, some of its neighbors might be colored, some might not. But no matter what, that vertex and its neighbors could all be colored distinctly, since there are at most ∆(G) neighbors, plus the one vertex being considered. In fact, there are examples of graphs for which χ(G) ∆(G) + 1. For any n, the complete graph K n has chromatic number n, but ∆(K n ) n − 1 (since every vertex is adjacent to every other vertex). Additionally, any odd cycle will have chromatic number 3, but the degree of every vertex in a cycle is 2. It turns out that these are the only two types of examples where we get equality, a result known as Brooks’ Theorem. Theorem 4.4.5 Brooks’ Theorem. Any graph G satisﬁes χ(G) ≤ ∆(G), unless G is a complete graph or an odd cycle, in which case χ(G) ∆(G) + 1. The proof of this theorem is just complicated enough that we will not present it here (although you are asked to prove a special case in the exercises). The adventurous reader is encouraged to ﬁnd a book on graph theory for suggestions on how to prove the theorem. Coloring Edges The chromatic number of a graph tells us about coloring vertices, but we could also ask about coloring edges. Just like with vertex coloring, we might insist that edges that are adjacent must be colored diﬀerently. Here, we are thinking of two edges as being adjacent if they are incident to the same vertex. The least number of colors required to properly color the edges of a graph G is called the chromatic index of G, written χ′(G) . Example 4.4.6 Six friends decide to spend the afternoon playing chess. Everyone will play everyone else once. They have plenty of chess sets but nobody wants to play more than one game at a time. Games will last an hour (thanks to their handy chess clocks). How many hours will the tournament last? Solution. Represent each player with a vertex and put an edge between two players if they will play each other. In this case, we get the graph K6 : 4.4. Coloring 273 We must color the edges; each color represents a diﬀerent hour. Since diﬀerent edges incident to the same vertex will be colored diﬀerently, no player will be playing two diﬀerent games (edges) at the same time. Thus we need to know the chromatic index of K 6 . Notice that for sure χ′(K 6 ) ≥ 5, since there is a vertex of degree 5. It turns out 5 colors is enough (go ﬁnd such a coloring). Therefore the friends will play for 5 hours. Interestingly, if one of the friends in the above example left, the remaining 5 chess-letes would still need 5 hours: the chromatic index of K 5 is also 5. In general, what can we say about chromatic index? Certainly χ′(G) ≥ ∆(G). But how much higher could it be? Only a little higher. Theorem 4.4.7 Vizing’s Theorem. χ′(G) is either ∆(G) or ∆(G) + 1. For any graph G, the chromatic index At ﬁrst this theorem makes it seem like chromatic index might not be very interesting. However, deciding which case a graph is in is not always easy. Graphs for which χ′(G) ∆(G) are called class 1, while the others are called class 2. Bipartite graphs always satisfy χ′(G) ∆(G), so are class 1 (this was proved by König in 1916, decades before Vizing proved his theorem in 1964). In 1965 Vizing proved that all planar graphs with ∆(G) ≥ 8 are of class 1, but this does not hold for all planar graphs with 2 ≤ ∆(G) ≤ 5. Vizing conjectured that all planar graphs with ∆(G) 6 or ∆(G) 7 are class 1; the ∆(G) 7 case was proved in 2001 by Sanders and Zhao; the ∆(G) 6 case is still open. Ramsey Theory. There is another interesting way we might consider coloring edges, quite diﬀerent from what we have discussed so far. What if we colored every edge of a graph either red or blue. Can we do so without, say, creating a monochromatic triangle (i.e., an all red or all blue triangle)? Certainly for some graphs the answer is yes. Try doing so for K4 . What about K 5 ? K 6 ? How far can we go? The problem above is not too diﬃcult and is a fun exercise. We could extend the question in a variety of ways. What if we had three colors? What if we were trying to avoid other graphs. Suprisingly, very little is known about these questions. For example, we know that you need to go up to K 17 in order to force a monochromatic triangle using three colors, but nobody knows how big you need to go with more colors. Similarly, we know that using two colors K 18 is the smallest graph that forces a monochromatic copy of K4 , but the best we have to force a monochromatic 274 4. Graph Theory K5 is a range, somewhere from K 43 to K49 . If you are interested in these sorts of questions, this area of graph theory is called Ramsey theory. Check it out. Exercises 1. What is the smallest number of colors you need to properly color the vertices of K4,5 ? That is, ﬁnd the chromatic number of the graph. 2. Draw a graph with chromatic number 6 (i.e., which requires 6 colors to properly color the vertices). Could your graph be planar? Explain. 3. Find the chromatic number of each of the following graphs. 4. A group of 10 friends decides to head up to a cabin in the woods (where nothing could possibly go wrong). Unfortunately, a number of these friends have dated each other in the past, and things are still a little awkward. To get to the cabin, they need to divide up into some number of cars, and no two people who dated should be in the same car. (a) What is the smallest number of cars you need if all the relationships were strictly heterosexual? Represent an example of such a situation with a graph. What kind of graph do you get? (b) Because a number of these friends dated there are also conﬂicts between friends of the same gender, listed below. Now what is the smallest number of conﬂict-free cars they could take to the cabin? Friend Conﬂicts A CFJ B J C AEF D H E CFG F ACEGI G EFI H D I AFG 5. What is the smallest number of colors that can be used to color the vertices of a cube so that no two adjacent vertices are colored identically? 6. Prove the chromatic number of any tree is two. Recall, a tree is a connected graph with no cycles. (a) Describe a procedure to color the tree below. J B 4.4. Coloring 275 (b) The chromatic number of C n is two when n is even. What goes wrong when n is odd? (c) Prove that your procedure from part (a) always works for any tree. (d) Now, prove using induction that every tree has chromatic number 2. 7. The two problems below can be solved using graph coloring. For each problem, represent the situation with a graph, say whether you should be coloring vertices or edges and why, and use the coloring to solve the problem. (a) Your Quidditch league has 5 teams. You will play a tournament next week in which every team will play every other team once. Each team can play at most one match each day, but there is plenty of time in the day for multiple matches. What is the fewest number of days over which the tournament can take place? (b) Ten members of Math Club are driving to a math conference in a neighboring state. However, some of these students have dated in the past, and things are still a little awkward. Each student lists which other students they refuse to share a car with; these conﬂicts are recored in the table below. What is the fewest number of cars the club needs to make the trip? Do not worry about running out of seats, just avoid the conﬂicts. Student Conﬂicts A BEJ B ADG C HJ D BF E AI F DJ G B H CI I EHJ J ACFI 8. Prove the 6-color theorem: every planar graph has chromatic number 6 or less. Do not assume the 4-color theorem (whose proof is MUCH harder), but you may assume the fact that every planar graph contains a vertex of degree at most 5. 9. Not all graphs are perfect. Give an example of a graph with chromatic number 4 that does not contain a copy of K4 . That is, there should be no 4 vertices all pairwise adjacent. 276 4. Graph Theory 10. Find the chromatic number of the graph below and prove you are correct. 11. Prove by induction on vertices that any graph G which contains at least one vertex of degree less than ∆(G) (the maximal degree of all vertices in G) has chromatic number at most ∆(G). 12. You have a set of magnetic alphabet letters (one of each of the 26 letters in the alphabet) that you need to put into boxes. For obvious reasons, you don’t want to put two consecutive letters in the same box. What is the fewest number of boxes you need (assuming the boxes are able to hold as many letters as they need to)? 13. Suppose you colored edges of a graph either red or blue (not requiring that adjacent edges be colored diﬀerently). What must be true of the graph to guarantee some vertex is incident to three edges of the same color? Prove your answer. 14. Prove that if you color every edge of K6 either red or blue, you are guaranteed a monochromatic triangle (that is, an all red or an all blue triangle). 4.5. Euler Paths and Circuits 4.5 277 Euler Paths and Circuits Investigate! An Euler path, in a graph or multigraph, is a walk through the graph which uses every edge exactly once. An Euler circuit is an Euler path which starts and stops at the same vertex. Our goal is to ﬁnd a quick way to check whether a graph (or multigraph) has an Euler path or circuit. 1. Which of the graphs below have Euler paths? Which have Euler circuits? 2. List the degrees of each vertex of the graphs above. Is there a connection between degrees and the existence of Euler paths and circuits? 3. Is it possible for a graph with a degree 1 vertex to have an Euler circuit? If so, draw one. If not, explain why not. What about an Euler path? 4. What if every vertex of the graph has degree 2. Is there an Euler path? An Euler circuit? Draw some graphs. 5. Below is part of a graph. Even though you can only see some of the vertices, can you deduce whether the graph will have an Euler path or circuit? ! Attempt the above activity before proceeding ! If we start at a vertex and trace along edges to get to other vertices, we create a walk through the graph. More precisely, a walk in a graph is a sequence of vertices such that every vertex in the sequence is adjacent to the vertices before and after it in the sequence. If the walk travels along every edge exactly once, then the walk is called an Euler path (or Euler walk). If, in addition, the starting and ending vertices are the same (so you trace along every edge exactly once and end up where you started), 278 4. Graph Theory then the walk is called an Euler circuit (or Euler tour). Of course if a graph is not connected, there is no hope of ﬁnding such a path or circuit. For the rest of this section, assume all the graphs discussed are connected. The bridges of Königsberg problem is really a question about the existence of Euler paths. There will be a route that crosses every bridge exactly once if and only if the graph below has an Euler path: This graph is small enough that we could actually check every possible walk that does not reuse edges, and in doing so convince ourselves that there is no Euler path (let alone an Euler circuit). On small graphs which do have an Euler path, it is usually not diﬃcult to ﬁnd one. Our goal is to ﬁnd a quick way to check whether a graph has an Euler path or circuit, even if the graph is quite large. One way to guarantee that a graph does not have an Euler circuit is to include a “spike,” a vertex of degree 1. a The vertex a has degree 1, and if you try to make an Euler circuit, you see that you will get stuck at the vertex. It is a dead end. That is, unless you start there. But then there is no way to return, so there is no hope of ﬁnding an Euler circuit. There is however an Euler path. It starts at the vertex a, then loops around the triangle. You will end at the vertex of degree 3. You run into a similar problem whenever you have a vertex of any odd degree. If you start at such a vertex, you will not be able to end there (after traversing every edge exactly once). After using one edge to leave the starting vertex, you will be left with an even number of edges emanating from the vertex. Half of these could be used for returning to the vertex, the other half for leaving. So you return, then leave. Return, then leave. The only way to use up all the edges is to use the last one by leaving the vertex. On the other hand, if you have a vertex with odd degree that you do not start a path at, then you will eventually get stuck at that vertex. The path will use pairs of edges incident to the vertex to arrive and leave again. Eventually all but one of these edges will be used up, leaving only an edge to arrive by, and none to leave again. What all this says is that if a graph has an Euler path and two vertices with odd degree, then the Euler path must start at one of the odd degree vertices and end at the other. In such a situation, every other vertex must 4.5. Euler Paths and Circuits 279 have an even degree since we need an equal number of edges to get to those vertices as to leave them. How could we have an Euler circuit? The graph could not have any odd degree vertex as an Euler path would have to start there or end there, but not both. Thus for a graph to have an Euler circuit, all vertices must have even degree. The converse is also true: if all the vertices of a graph have even degree, then the graph has an Euler circuit, and if there are exactly two vertices with odd degree, the graph has an Euler path. To prove this is a little tricky, but the basic idea is that you will never get stuck because there is an “outbound” edge for every “inbound” edge at every vertex. If you try to make an Euler path and miss some edges, you will always be able to “splice in” a circuit using the edges you previously missed. Euler Paths and Circuits. • A graph has an Euler circuit if and only if the degree of every vertex is even. • A graph has an Euler path if and only if there are at most two vertices with odd degree. Since the bridges of Königsberg graph has all four vertices with odd degree, there is no Euler path through the graph. Thus there is no way for the townspeople to cross every bridge exactly once. Hamilton Paths Suppose you wanted to tour Königsberg in such a way that you visit each land mass (the two islands and both banks) exactly once. This can be done. In graph theory terms, we are asking whether there is a path which visits every vertex exactly once. Such a path is called a Hamilton path (or Hamiltonian path). We could also consider Hamilton cycles, which are Hamliton paths which start and stop at the same vertex. Example 4.5.1 Determine whether the graphs below have a Hamilton path. Solution. The graph on the left has a Hamilton path (many diﬀerent ones, actually), as shown here: 280 4. Graph Theory The graph on the right does not have a Hamilton path. You would need to visit each of the “outside” vertices, but as soon as you visit one, you get stuck. Note that this graph does not have an Euler path, although there are graphs with Euler paths but no Hamilton paths. It appears that ﬁnding Hamilton paths would be easier because graphs often have more edges than vertices, so there are fewer requirements to be met. However, nobody knows whether this is true. There is no known simple test for whether a graph has a Hamilton path. For small graphs this is not a problem, but as the size of the graph grows, it gets harder and harder to check wither there is a Hamilton path. In fact, this is an example of a question which as far as we know is too diﬃcult for computers to solve; it is an example of a problem which is NP-complete. Exercises 1. You and your friends want to tour the southwest by car. You will visit the nine states below, with the following rather odd rule: you must cross each border between neighboring states exactly once (so, for example, you must cross the Colorado-Utah border exactly once). Can you do it? If so, does it matter where you start your road trip? What fact about graph theory solves this problem? 2. Which of the following graphs contain an Euler path? Which contain an Euler circuit? (a) K 4 (b) K 5 . (c) K5,7 (d) K2,7 (e) C 7 (f) P7 3. Edward A. Mouse has just ﬁnished his brand new house. The ﬂoor plan is shown below: 4.5. Euler Paths and Circuits 281 (a) Edward wants to give a tour of his new pad to a lady-mousefriend. Is it possible for them to walk through every doorway exactly once? If so, in which rooms must they begin and end the tour? Explain. (b) Is it possible to tour the house visiting each room exactly once (not necessarily using every doorway)? Explain. (c) After a few mouse-years, Edward decides to remodel. He would like to add some new doors between the rooms he has. Of course, he cannot add any doors to the exterior of the house. Is it possible for each room to have an odd number of doors? Explain. 4. For which n does the graph K n contain an Euler circuit? Explain. 5. For which m and n does the graph K m,n contain an Euler path? An Euler circuit? Explain. 6. For which n does K n contain a Hamilton path? A Hamilton cycle? Explain. 7. For which m and n does the graph K m,n contain a Hamilton path? A Hamilton cycle? Explain. 8. A bridge builder has come to Königsberg and would like to add bridges so that it is possible to travel over every bridge exactly once. How many bridges must be built? 9. Below is a graph representing friendships between a group of students (each vertex is a student and each edge is a friendship). Is it possible for the students to sit around a round table in such a way that every student sits between two friends? What does this question have to do with paths? 282 4. Graph Theory 10. On the table rest 8 dominoes, as shown below. If you were to line them up in a single row, so that any two sides touching had matching numbers, what would the sum of the two end numbers be? 11. Is there anything we can say about whether a graph has a Hamilton path based on the degrees of its vertices? (a) Suppose a graph has a Hamilton path. What is the maximum number of vertices of degree one the graph can have? Explain why your answer is correct. (b) Find a graph which does not have a Hamilton path even though no vertex has degree one. Explain why your example works. 12. Consider the following graph: (a) Find a Hamilton path. Can your path be extended to a Hamilton cycle? (b) Is the graph bipartite? If so, how many vertices are in each “part”? (c) Use your answer to part (b) to prove that the graph has no Hamilton cycle. (d) Suppose you have a bipartite graph G in which one part has at least two more vertices than the other. Prove that G does not have a Hamilton path. 4.6. Matching in Bipartite Graphs 4.6 283 Matching in Bipartite Graphs Investigate! Given a bipartite graph, a matching is a subset of the edges for which every vertex belongs to exactly one of the edges. Our goal in this activity is to discover some criterion for when a bipartite graph has a matching. Does the graph below contain a matching? If so, ﬁnd one. Not all bipartite graphs have matchings. Draw as many fundamentally diﬀerent examples of bipartite graphs which do NOT have matchings. Your goal is to ﬁnd all the possible obstructions to a graph having a perfect matching. Write down the necessary conditions for a graph to have a matching (that is, ﬁll in the blank: If a graph has a matching, then ). Then ask yourself whether , then the these conditions are suﬃcient (is it true that if graph has a matching?). ! Attempt the above activity before proceeding ! We conclude with one more example of a graph theory problem to illustrate the variety and vastness of the subject. Suppose you have a bipartite graph G. This will consist of two sets of vertices A and B with some edges connecting some vertices of A to some vertices in B (but of course, no edges between two vertices both in A or both in B). A matching of A is a subset of the edges for which each vertex of A belongs to exactly one edge of the subset, and no vertex in B belongs to more than one edge in the subset. In practice we will assume that |A| |B| (the two sets have the same number of vertices) so this says that every vertex in the graph belongs to exactly one edge in the matching.9 Some context might make this easier to understand. Think of the vertices in A as representing students in a class, and the vertices in B as representing presentation topics. We put an edge from a vertex a ∈ A to a vertex b ∈ B if student a would like to present on topic b. Of course, some students would want to present on more than one topic, so their vertex would have degree greater than 1. As the teacher, you want to assign each 9Note: what we are calling a matching is sometimes called a perfect matching or complete matching. This is because in it interesting to look at non-perfect matchings as well. We will call those partial matchings. 284 4. Graph Theory student their own unique topic. Thus you want to ﬁnd a matching of A: you pick some subset of the edges so that each student gets matched up with exactly one topic, and no topic gets matched to two students.10 The question is: when does a bipartite graph contain a matching of A? To begin to answer this question, consider what could prevent the graph from containing a matching. This will not necessarily tell us a condition when the graph does have a matching, but at least it is a start. One way G could not have a matching is if there is a vertex in A not adjacent to any vertex in B (so having degree 0). What else? What if two students both like the same one topic, and no others? Then after assigning that one topic to the ﬁrst student, there is nothing left for the second student to like, so it is very much as if the second student has degree 0. Or what if three students like only two topics between them. Again, after assigning one student a topic, we reduce this down to the previous case of two students liking only one topic. We can continue this way with more and more students. It should be clear at this point that if there is every a group of n students who as a group like n − 1 or fewer topics, then no matching is possible. This is true for any value of n, and any group of n students. To make this more graph-theoretic, say you have a set S ⊆ A of vertices. Deﬁne N(S) to be the set of all the neighbors of vertices in S. That is, N(S) contains all the vertices (in B) which are adjacent to at least one of the vertices in S. (In the student/topic graph, N(S) is the set of topics liked by the students of S.) Our discussion above can be summarized as follows: Matching Condition. If a bipartite graph G {A, B} has a matching of A, then |N(S)| ≥ |S| for all S ⊆ A. Is the converse true? Suppose G satisﬁes the matching condition |N(S)| ≥ |S| for all S ⊆ A (every set of vertices has at least as many neighbors than vertices in the set). Does that mean that there is a matching? 10The standard example for matchings used to be the marriage problem in which A consisted of the men in the town, B the women, and an edge represented a marriage that was agreeable to both parties. A matching then represented a way for the town elders to marry oﬀ everyone in the town, no polygamy allowed. We have chosen a more progressive context for the sake of political correctness. 4.6. Matching in Bipartite Graphs 285 Surprisingly, yes. The obvious necessary condition is also suﬃcient.11 This is a theorem ﬁrst proved by Philip Hall in 1935.12 Theorem 4.6.1 Hall’s Marriage Theorem. Let G be a bipartite graph with sets A and B. Then G has a matching of A if and only if |N(S)| ≥ |S| for all S ⊆ A. There are quite a few diﬀerent proofs of this theorem – a quick internet search will get you started. In addition to its application to marriage and student presentation topics, matchings have applications all over the place. We conclude with one such example. Example 4.6.2 Suppose you deal 52 regular playing cards into 13 piles of 4 cards each. Prove that you can always select one card from each pile to get one of each of the 13 card values Ace, 2, 3, . . . , 10, Jack, Queen, and King. Solution. Doing this directly would be diﬃcult, but we can use the matching condition to help. Construct a graph G with 13 vertices in the set A, each representing one of the 13 card values, and 13 vertices in the set B, each representing one of the 13 piles. Draw an edge between a vertex a ∈ A to a vertex b ∈ B if a card with value a is in the pile b. Notice that we are just looking for a matching of A; each value needs to be found in the piles exactly once. We will have a matching if the matching condition holds. Given any set of card values (a set S ⊆ A) we must show that |N(S)| ≥ |S|. That is, the number of piles that contain those values is at least the number of diﬀerent values. But what if it wasn’t? Say |S| k. If |N(S)| < k, then we would have fewer than 4k diﬀerent cards in those piles (since each pile contains 4 cards). But there are 4k cards with the k diﬀerent values, so at least one of these cards must be in another pile, a contradiction. Thus the matching condition holds, so there is a matching, as required. 11This happens often in graph theory. If you can avoid the obvious counterexamples, you often get what you want. 12There is also an inﬁnite version of the theorem which was proved by Marshal Hall, Jr. The name is a coincidence though as the two Halls are not related. 286 4. Graph Theory Exercises 1. Find a matching of the bipartite graphs below or explain why no matching exists. 2. A bipartite graph that doesn’t have a matching might still have a partial matching. By this we mean a set of edges for which no vertex belongs to more than one edge (but possibly belongs to none). Every bipartite graph (with at least one edge) has a partial matching, so we can look for the largest partial matching in a graph. Your “friend” claims that she has found the largest partial matching for the graph below (her matching is in bold). She explains that no other edge can be added, because all the edges not used in her partial matching are connected to matched vertices. Is she correct? 3. One way you might check to see whether a partial matching is maximal is to construct an alternating path. This is a sequence of adjacent edges, which alternate between edges in the matching and edges not in the matching (no edge can be used more than once). If an alternating path starts and stops with an edge not in the matching, then it is called an augmenting path. (a) Find the largest possible alternating path for the partial matching of your friend’s graph. Is it an augmenting path? How would this help you ﬁnd a larger matching? (b) Find the largest possible alternating path for the partial matching below. Are there any augmenting paths? Is the partial matching the largest one that exists in the graph? 4.6. Matching in Bipartite Graphs 4. 287 The two richest families in Westeros have decided to enter into an alliance by marriage. The ﬁrst family has 10 sons, the second has 10 girls. The ages of the kids in the two families match up. To avoid impropriety, the families insist that each child must marry someone either their own age, or someone one position younger or older. In fact, the graph representing agreeable marriages looks like this: The question: how many diﬀerent acceptable marriage arrangements which marry oﬀ all 20 children are possible? (a) How many marriage arrangements are possible if we insist that there are exactly 6 boys marry girls not their own age? (b) Could you generalize the previous answer to arrive at the total number of marriage arrangements? (c) How do you know you are correct? Try counting in a diﬀerent way. Look at smaller family sizes and get a sequence. (d) Can you give a recurrence relation that ﬁts the problem? 5. We say that a set of vertices A ⊆ V is a vertex cover if every edge of the graph is incident to a vertex in the cover (so a vertex cover covers the edges). Since V itself is a vertex cover, every graph has a vertex cover. The interesting question is about ﬁnding a minimal vertex cover, one that uses the fewest possible number of vertices. (a) Suppose you had a matching of a graph. How can you use that to get a minimal vertex cover? Will your method always work? (b) Suppose you had a minimal vertex cover for a graph. How can you use that to get a partial matching? Will your method always work? (c) What is the relationship between the size of the minimal vertex cover and the size of the maximal partial matching in a graph? 288 6. 4. Graph Theory For many applications of matchings, it makes sense to use bipartite graphs. You might wonder, however, whether there is a way to ﬁnd matchings in graphs in general. (a) For which n does the complete graph K n have a matching? (b) Prove that if a graph has a matching, then |V | is even. (c) Is the converse true? That is, do all graphs with |V | even have a matching? (d) What if we also require the matching condition? Prove or disprove: If a graph with an even number of vertices satisﬁes |N(S)| ≥ |S| for all S ⊆ V, then the graph has a matching. 4.7. Chapter Summary 4.7 289 Chapter Summary Hopefully this chapter has given you some sense for the wide variety of graph theory topics as well as why these studies are interesting. There are many more interesting areas to consider and the list is increasing all the time; graph theory is an active area of mathematical research. One reason graph theory is such a rich area of study is that it deals with such a fundamental concept: any pair of objects can either be related or not related. What the objects are and what “related” means varies on context, and this leads to many applications of graph theory to science and other areas of math. The objects can be countries, and two countries can be related if they share a border. The objects could be land masses which are related if there is a bridge between them. The objects could be websites which are related if there is a link from one to the other. Or we can be completely abstract: the objects are vertices which are related if their is an edge between them. What question we ask about the graph depends on the application, but often leads to deeper, general and abstract questions worth studying in their own right. Here is a short summary of the types of questions we have considered: • Can the graph be drawn in the plane without edges crossing? If so, how many regions does this drawing divide the plane into? • Is it possible to color the vertices of the graph so that related vertices have diﬀerent colors using a small number of colors? How many colors are needed? • Is it possible to trace over every edge of a graph exactly once without lifting up your pencil? What other sorts of “paths” might a graph posses? • Can you ﬁnd subgraphs with certain properties? For example, when does a (bipartite) graph contain a subgraph in which all vertices are only related to one other vertex? Not surprisingly, these questions are often related to each other. For example, the chromatic number of a graph cannot be greater than 4 when the graph is planar. Whether the graph has an Euler path depends on how many vertices each vertex is adjacent to (and whether those numbers are 290 4. Graph Theory always even or not). Even the existence of matchings in bipartite graphs can be proved using paths. Chapter Review 1. Which (if any) of the graphs below are the same? Which are diﬀerent? Explain. 2. Which of the graphs in the previous question contain Euler paths or circuits? Which of the graphs are planar? 3. Draw a graph which has an Euler circuit but is not planar. 4. Draw a graph which does not have an Euler path and is also not planar. 5. Consider the graph G (V, E) with V {a, b, c, d, e , f , g} and E {ab, ac, a f , b g, cd, ce} (here we are using the shorthand for edges: ab really means {a, b}, for example). (a) Is the graph G isomorphic to G′ (V ′ , E′) with V ′ {t, u, v, w, x, y, z} and E′ {tz, uv, u y, uz, vw, vx}? If so, give the isomoprhism. If not, explain how you know. (b) Find a graph G′′ with 7 vertices and 6 edges which is NOT isomorphic to G, or explain why this is not possible. (c) Write down the degree sequence for G. That is, write down the degrees of all the vertices, in decreasing order. (d) Find a connected graph G′′′ with the same degree sequence of G which is NOT isomorphic to G, or explain why this is not possible. (e) What kind of graph is G? Is G complete? Bipartite? A tree? A cycle? A path? A wheel? (f) Is G planar? (g) What is the chromatic number of G? Explain. (h) Does G have an Euler path or circuit? Explain. 6. If a graph has 10 vertices and 10 edges and contains an Euler circuit, must it be planar? How many faces would it have? 4.7. Chapter Summary 7. 291 Suppose G is a graph with n vertices, each having degree 5. (a) For which values of n does this make sense? (b) For which values of n does the graph have an Euler path? (c) What is the smallest value of n for which the graph might be planar? (tricky) 8. At a school dance, 6 girls and 4 boys take turns dancing (as couples) with each other. (a) How many couples danced if every girl dances with every boy? (b) How many couples danced if everyone danced with everyone else (regardless of gender)? (c) Explain what graphs can be used to represent these situations. 9. Among a group of n people, is it possible for everyone to be friends with an odd number of people in the group? If so, what can you say about n? 10. Your friend has challenged you to create a convex polyhedron containing 9 triangles and 6 pentagons. (a) Is it possible to build such a polyhedron using only these shapes? Explain. (b) You decide to also include one heptagon (seven-sided polygon). How many vertices does your new convex polyhedron contain? (c) Assuming you are successful in building your new 16-faced polyhedron, could every vertex be the joining of the same number of faces? Could each vertex join either 3 or 4 faces? If so, how many of each type of vertex would there be? 11. Is there a convex polyhedron which requires 5 colors to properly color the vertices of the polyhedron? Explain. 12. How many edges does the graph K n,n have? For which values of n does the graph contain an Euler circuit? For which values of n is the graph planar? 13. The graph G has 6 vertices with degrees 1, 2, 2, 3, 3, 5. How many edges does G have? If G was planar how many faces would it have? Does G have an Euler path? 14. What is the smallest number of colors you need to properly color the vertices of K 7 . Can you say whether K 7 is planar based on your answer? 292 4. Graph Theory 15. What is the smallest number of colors you need to properly color the vertices of K3,4 ? Can you say whether K 3,4 is planar based on your answer? 16. Prove that K3,4 is not planar. Do this using Euler’s formula, not just by appealing to the fact that K 3,3 is not planar. 17. A dodecahedron is a regular convex polyhedron made up of 12 regular pentagons. (a) Suppose you color each pentagon with one of three colors. Prove that there must be two adjacent pentagons colored identically. (b) What if you use four colors? (c) What if instead of a dodecahedron you colored the faces of a cube? 18. Decide whether the following statements are true or false. Prove your answers. (a) If two graph G1 and G2 have the same chromatic number, then they are isomorphic. (b) If two graphs G1 and G2 have the same number of vertices and edges and have the same chromatic number, then they are isomorphic. (c) If two graphs are isomorphic, then they have the same chromatic number. 19. If a planar graph G with 7 vertices divides the plane into 8 regions, how many edges must G have? 20. Consider the graph below: (a) Does the graph have an Euler path or circuit? Explain. (b) Is the graph planar? Explain. (c) Is the graph bipartite? Complete? Complete bipartite? (d) What is the chromatic number of the graph. 4.7. Chapter Summary 293 21. For each part below, say whether the statement is true or false. Explain why the true statements are true, and give counterexamples for the false statements. (a) Every bipartite graph is planar. (b) Every bipartite graph has chromatic number 2. (c) Every bipartite graph has an Euler path. (d) Every vertex of a bipartite graph has even degree. (e) A graph is bipartite if and only if the sum of the degrees of all the vertices is even. 22. Consider the statement “If a graph is planar, then it has an Euler path.” (a) Write the converse of the statement. (b) Write the contrapositive of the statement. (c) Write the negation of the statement. (d) Is it possible for the contrapositive to be false? If it was, what would that tell you? (e) Is the original statement true or false? Prove your answer. (f) Is the converse of the statement true or false? Prove your answer. 23. Let G be a connected graph with v vertices and e edges. Use mathematical induction to prove that if G contains exactly one cycle (among other edges and vertices), then v e. Note: this is asking you to prove a special case of Euler’s formula for planar graphs, so do not use that formula in your proof. 294 4. Graph Theory Chapter 5 Additional Topics 5.1 Generating Functions There is an extremely powerful tool in discrete mathematics used to manipulate sequences called the generating function. The idea is this: instead of an inﬁnite sequence (for example: 2, 3, 5, 8, 12, . . .) we look at a single function which encodes the sequence. But not a function which gives the nth term as output. Instead, a function whose power series (like from calculus) “displays” the terms of the sequence. So for example, we would look at the power series 2+3x +5x 2 +8x 3 +12x 4 +· · · which displays the sequence 2, 3, 5, 8, 12, . . . as coeﬃcients. An inﬁnite power series is simply an inﬁnite sum of terms of the form c n x n were c n is some constant. So we might write a power series like this: ∞ Õ ck x k . k0 or expanded like this c0 + c1 x + c2 x 2 + c3 x 3 + c4 x 4 + c5 x 5 + · · · . When viewed in the context of generating functions, we call such a power series a generating series. The generating series generates the sequence c0 , c1 , c2 , c3 , c4 , c5 , . . . . In other words, the sequence generated by a generating series is simply the sequence of coeﬃcients of the inﬁnite polynomial. Example 5.1.1 What sequence is represented by the generating series 3 + 8x 2 + 5 x 3 + x7 + 100x 6 + · · ·? Solution. We just read oﬀ the coeﬃcients of each x n term. So a 0 3 since the coeﬃcient of x 0 is 3 (x 0 1 so this is the constant term). What is a 1 ? It is NOT 8, since 8 is the coeﬃcient of x 2 , so 8 is the term a 2 of the sequence. To ﬁnd a 1 we need to look for the coeﬃcient of x 1 which in this case is 0. So a 1 0. Continuing, we 295 296 5. Additional Topics have a 2 8, a 3 1, a 4 0, and a 5 17 . So we have the sequence 1 3, 0, 8, 1, , 100, . . . . 7 Note that when discussing generating functions, we always start our sequence with a 0 . Now you might very naturally ask why we would do such a thing. One reason is that encoding a sequence with a power series helps us keep track of which term is which in the sequence. For example, if we write the sequence 1, 3, 4, 6, 9, . . . , 24, 41, . . . it is impossible to determine which term 24 is (even if we agreed that the ﬁrst term was supposed to be a 0 ). However, if we wrote the generating series instead, we would have 1+3x +4x 2 +6x 3 +9x 4 +· · ·+24x 17 +41x 18 +· · ·. Now it is clear that 24 is the 17th term of the sequence (that is, a 17 24). Of course to get this beneﬁt we could have displayed our sequence in any number of ways, perhaps 1 0 3 1 4 2 6 3 9 4 · · · 24 17 41 18 · · ·, but we do not do this. The reason is that the generating series looks like an ordinary power series (although we are interpreting it diﬀerently) so we can do things with it that we ordinarily do with power series such as write down what it converges to. 2 For example, from calculus we know that the power series 1 + x + x2 + x4 xn x3 x x 6 + 24 + · · · + n! + · · · converges to the function e . So we can use e as a way of talking about the sequence of coeﬃcients of the power series for e x . When we write down a nice compact function which has an inﬁnite power series that we view as a generating series, then we call that function a generating function. In this example, we would say 1 1 1 1 1, 1, , , , . . . , , . . . has generating function e x . 2 6 24 n! Building Generating Functions The e x example is very speciﬁc. We have a rather odd sequence, and the only reason we know its generating function is because we happen to know the Taylor series for e x . Our goal now is to gather some tools to build the generating function of a particular given sequence. Let’s see what the generating functions are for some very simple sequences. The simplest of all: 1, 1, 1, 1, 1, . . . . What does the generating series look like? It is simply 1 + x + x 2 + x 3 + x 4 + · · ·. Now, can we ﬁnd a closed formula for this power series? Yes! This particular series is really just a geometric series with common ratio x. So if we use our “multiply, 5.1. Generating Functions 297 shift and subtract” technique from Section 2.2, we have S 1 + x + x2 + x3 + · · · −xS x + x2 + x3 + x4 + · · · (1 − x)S 1 Therefore we see that 1 + x + x2 + x3 · · · 1 . 1−x You might remember from calculus that this is only true on the interval of convergence for the power series, in this case when |x| < 1. That is true for us, but we don’t care. We are never going to plug anything in for x, so as long as there is some value of x for which the generating function and generating series agree, we are happy. And in this case we are happy. 1, 1, 1, . . .. The generating function for 1, 1, 1, 1, 1, 1, . . . is 1 1−x Let’s use this basic generating function to ﬁnd generating functions for more sequences. What if we replace x by −x. We get 1 1 − x + x 2 − x 3 + · · · which generates 1, −1, 1, −1, . . . . 1+x If we replace x by 3x we get 1 1 + 3x + 9x 2 + 27x 3 + · · · which generates 1, 3, 9, 27, . . . . 1 − 3x 1 By replacing the x in 1−x we can get generating functions for a variety of sequences, but not all. For example, you cannot plug in anything for x to get the generating function for 2, 2, 2, 2, . . .. However, we are not lost yet. Notice that each term of 2, 2, 2, 2, . . . is the result of multiplying the terms of 1, 1, 1, 1, . . . by the constant 2. So multiply the generating function by 2 as well. 2 2 + 2x + 2x 2 + 2x 3 + · · · which generates 2, 2, 2, 2, . . . . 1−x Similarly, to ﬁnd the generating function for the sequence 3, 9, 27, 81, . . ., we note that this sequence is the result of multiplying each term of 1, 3, 9, 27, . . . by 3. Since we have the generating function for 1, 3, 9, 27, . . . we can say 3 3 · 1 + 3 · 3x + 3 · 9x 2 + 3 · 27x 3 + · · · which generates 3, 9, 27, 81, . . . . 1 − 3x 298 5. Additional Topics What about the sequence 2, 4, 10, 28, 82, . . .? Here the terms are always 1 more than powers of 3. That is, we have added the sequences 1, 1, 1, 1, . . . and 1, 3, 9, 27, . . . term by term. Therefore we can get a generating function by adding the respective generating functions: 2 + 4x + 10x 2 + 28x 3 + · · · (1 + 1) + (1 + 3)x + (1 + 9)x 2 + (1 + 27)x 3 + · · · 1 + x + x 2 + x 3 + · · · + 1 + 3x + 9x 2 + 27x 3 + · · · 1 1 + 1 − x 1 − 3x The fun does not stop there: if we replace x in our original generating function by x 2 we get 1 1 + x 2 + x 4 + x 6 · · · which generates 1, 0, 1, 0, 1, 0, . . . . 1 − x2 How could we get 0, 1, 0, 1, 0, 1, . . .? Start with the previous sequence and shift it over by 1. But how do you do this? To see how shifting works, let’s ﬁrst try to get the generating function for the sequence 0, 1, 3, 9, 27, . . .. 1 1 + 3x + 9x 2 + 27x 3 + · · ·. To get the zero out front, We know that 1−3x we need the generating series to look like x + 3x 2 + 9x 3 + 27x 4 + · · · (so there is no constant term). Multiplying by x has this eﬀect. So the x . This will also work to get generating function for 0, 1, 3, 9, 27, . . . is 1−3x the generating function for 0, 1, 0, 1, 0, 1, . . .: x x + x 3 + x 5 + · · · which generates 0, 1, 0, 1, 0, 1 . . . . 1 − x2 What if we add the sequences 1, 0, 1, 0, 1, 0, . . . and 0, 1, 0, 1, 0, 1, . . . term by term? We should get 1, 1, 1, 1, 1, 1 . . .. What happens when we add the generating functions? It works (try it)! x 1 1 + . 2 2 1−x 1−x 1−x 1 Here’s a sneaky one: what happens if you take the derivative of 1−x ? 1 We get (1−x)2 . On the other hand, if we diﬀerentiate term by term in the power series, we get (1 + x + x 2 + x 3 + · · · )′ 1 + 2x + 3x 2 + 4x 3 + · · · which is the generating series for 1, 2, 3, 4, . . .. This says 1, 2, 3, . . .. The generating function for 1, 2, 3, 4, 5, . . . is 1 . (1 − x)2 5.1. Generating Functions Take a second derivative: 2 (1−x)3 2 + 6x + 12x 2 + 20x 3 + · · ·. So 299 1 (1−x)3 1+3x +6x 2 +10x 3 +· · · is a generating function for the triangular numbers, 1, 3, 6, 10 . . . (although here we have a 0 1 while T0 0 usually). Differencing 1 We have seen how to ﬁnd generating functions from 1−x using multiplication (by a constant or by x), substitution, addition, and diﬀerentiation. To use each of these, you must notice a way to transform the sequence 1, 1, 1, 1, 1 . . . into your desired sequence. This is not always easy. It is also not really the way we have analyzed sequences. One thing we have considered often is the sequence of diﬀerences between terms of a sequence. This will turn out to be helpful in ﬁnding generating functions as well. The sequence of diﬀerences is often simpler than the original sequence. So if we know a generating function for the diﬀerences, we would like to use this to ﬁnd a generating function for the original sequence. For example, consider the sequence 2, 4, 10, 28, 82, . . .. How could we move to the sequence of ﬁrst diﬀerences: 2, 6, 18, 54, . . .? We want to subtract 2 from the 4, 4 from the 10, 10 from the 28, and so on. So if we subtract (term by term) the sequence 0, 2, 4, 10, 28, . . . from 2, 4, 10, 28 . . ., we will be set. We can get the generating function for 0, 2, 4, 10, 28, . . . from the generating function for 2, 4, 10, 28 . . . by multiplying by x. Use A to represent the generating function for 2, 4, 10, 28, 82, . . . Then: A 2 + 4x + 10x 2 + 28x 3 + 82x 4 + · · · − xA 0 + 2x + 4x 2 + 10x 3 + 28x 4 + 82x 5 + · · · (1 − x)A 2 + 2x + 6x 2 + 18x 3 + 54x 4 + · · · While we don’t get exactly the sequence of diﬀerences, we do get something close. In this particular case, we already know the generating function A (we found it in the previous section) but most of the time we will use this diﬀerencing technique to ﬁnd A: if we have the generating function for the sequence of diﬀerences, we can then solve for A. Example 5.1.2 Find a generating function for 1, 3, 5, 7, 9, . . .. Solution. Notice that the sequence of diﬀerences is constant. We know how to ﬁnd the generating function for any constant sequence. So denote the generating function for 1, 3, 5, 7, 9, . . . by A. We have 300 5. Additional Topics A 1 + 3x + 5x 2 + 7x 3 + 9x 4 + · · · − xA 0 + x + 3x 2 + 5x 3 + 7x 4 + 9x 5 + · · · (1 − x)A 1 + 2x + 2x 2 + 2x 3 + 2x 4 + · · · We know that 2x + 2x 2 + 2x 3 + 2x 4 + · · · (1 − x)A 1 + 2x . Thus 1−x 2x . 1−x Now solve for A: A 2x 1 1+x + . 1 − x (1 − x)2 (1 − x)2 Does this makes sense? Before we simpliﬁed the two fractions into one, we were adding the generating function for the sequence 1, 1, 1, 1, . . . to the generating function for the sequence 1 0, 2, 4, 6, 8, 10, . . . (remember (1−x) 2 generates 1, 2, 3, 4, 5, . . ., multiplying by 2x shifts it over, putting the zero out front, and doubles each term). If we add these term by term, we get the correct sequence 1, 3, 5, 7, 9, . . .. Now that we have a generating function for the odd numbers, we can use that to ﬁnd the generating function for the squares: Example 5.1.3 Find the generating function for 1, 4, 9, 16, . . .. Note we take 1 a 0 . Solution. Again we call the generating function for the sequence A. Using diﬀerencing: A 1 + 4x + 9x 2 + 16x 3 + · · · − xA 0 + x + 4x 2 + 9x 3 + 16x 4 + · · · (1 − x)A 1 + 3x + 5x 2 + 7x 3 + · · · Since 1 + 3x + 5x 2 + 7x 3 + · · · 1+x 1+x we have A . (1 − x)2 (1 − x)3 In each of the examples above, we found the diﬀerence between consecutive terms which gave us a sequence of diﬀerences for which we knew a generating function. We can generalize this to more complicated 5.1. Generating Functions 301 relationships between terms of the sequence. For example, if we know that the sequence satisﬁes the recurrence relation a n 3a n−1 − 2a n−2 ? In other words, if we take a term of the sequence and subtract 3 times the previous term and then add 2 times the term before that, we get 0 (since a n − 3a n−1 + 2a n−2 0). That will hold for all but the ﬁrst two terms of the sequence. So after the ﬁrst two terms, the sequence of results of these calculations would be a sequence of 0’s, for which we deﬁnitely know a generating function. Example 5.1.4 The sequence 1, 3, 7, 15, 31, 63, . . . satisﬁes the recurrence relation a n 3a n−1 − 2a n−2 . Find the generating function for the sequence. Solution. Call the generating function for the sequence A. We have A 1 + 3x + 7x 2 + 15x 3 + 31x 4 + · · · + a n x n + · · · −3xA 0 − 3x − 9x 2 − 21x 3 − 45x 4 − · · · − 3a n−1 x n − · · · + 2x 2 A 0 + 0x + 2x 2 + 6x 3 + 14x 4 + · · · + 2a n−2 x n + · · · (1 − 3x + 2x 2 )A 1 We multiplied A by −3x which shifts every term over one spot and multiplies them by −3. On the third line, we multiplied A by 2x 2 , which shifted every term over two spots and multiplied them by 2. When we add up the corresponding terms, we are taking each term, subtracting 3 times the previous term, and adding 2 times the term before that. This will happen for each term after a 1 because a n − 3a n−1 + 2a n−2 0. In general, we might have two terms from the beginning of the generating series, although in this case the second term happens to be 0 as well. Now we just need to solve for A: A 1 . 1 − 3x + 2x 2 Multiplication and Partial Sums What happens to the sequences when you multiply two generating functions? Let’s see: A a 0 + a 1 x + a 2 x 2 + · · · and B b0 + b1 x + b2 x 2 + · · ·. To multiply A and B, we need to do a lot of distributing (inﬁnite FOIL?) but keep in mind we will group like terms and only need to write down the ﬁrst few terms to see the pattern. The constant term is a 0 b0 . The coeﬃcient of x is a 0 b1 + a 1 b0 . And so on. We get: AB a 0 b0 +(a 0 b1 +a1 b0 )x+(a 0 b 2 +a 1 b1 +a 2 b0 )x 2 +(a 0 b3 +a 1 b2 +a 2 b1 +a 3 b0 )x 3 +· · · . 302 5. Additional Topics Example 5.1.5 “Multiply” the sequence 1, 2, 3, 4, . . . by the sequence 1, 2, 4, 8, 16, . . .. Solution. The new constant term is just 1 · 1. The next term will be 1 · 2 + 2 · 1 4. The next term: 1 · 4 + 2 · 2 + 3 · 1 11. One more: 1 · 8 + 2 · 4 + 3 · 2 + 4 · 1 28. The resulting sequence is 1, 4, 11, 28, 57, . . . . Since the generating function for 1, 2, 3, 4, . . . is generating function for 1, 2, 4, 8, 16, . . . is generating function for 1, 4, 11, 28, 57, . . . is 1 1−2x , 1 (1−x)2 and the we have that the 1 (1−x)2 (1−2x) Consider the special case when you multiply a sequence by 1, 1, 1, . . .. For example, multiply 1, 1, 1, . . . by 1, 2, 3, 4, 5 . . .. The ﬁrst term is 1·1 1. Then 1 · 2 + 1 · 1 3. Then 1 · 3 + 1 · 2 + 1 · 1 6. The next term will be 10. We are getting the triangular numbers. More precisely, we get the sequence of partial sums of 1, 2, 3, 4, 5, . . .. In terms of generating functions, we 1 1 take 1−x (generating 1, 1, 1, 1, 1 . . .) and multiply it by (1−x) 2 (generating 1 1, 2, 3, 4, 5, . . .) and this give (1−x) 3 . This should not be a surprise as we found the same generating function for the triangular numbers earlier. The point is, if you need to ﬁnd a generating function for the sum of the ﬁrst n terms of a particular sequence, and you know the generating 1 function for that sequence, you can multiply it by 1−x . To go back from the sequence of partial sums to the original sequence, you look at the sequence of diﬀerences. When you get the sequence of diﬀerences you 1 end up multiplying by 1 − x, or equivalently, dividing by 1−x . Multiplying 1 1 by 1−x gives partial sums, dividing by 1−x gives diﬀerences. Solving Recurrence Relations with Generating Functions We conclude with an example of one of the many reasons studying generating functions is helpful. We can use generating functions to solve recurrence relations. Example 5.1.6 Solve the recurrence relation a n 3a n−1 − 2a n−2 with initial conditions a 0 1 and a 1 3. Solution. We saw in an example above that this recurrence relation gives the sequence 1, 3, 7, 15, 31, 63, . . . which has generating func1 . We did this by calling the generating function A tion 1 − 3x + 2x 2 5.1. Generating Functions 303 and then computing A − 3xA + 2x 2 A which was just 1, since every other term canceled out. But how does knowing the generating function help us? First, break up the generating function into two simpler ones. For this, we can use partial fraction decomposition. Start by factoring the denominator: 1 1 . 2 (1 − x)(1 − 2x) 1 − 3x + 2x Partial fraction decomposition tells us that we can write this faction as the sum of two fractions (we decompose the given fraction): 1 a b + (1 − x)(1 − 2x) 1 − x 1 − 2x for some constants a and b. To ﬁnd a and b we add the two decomposed fractions using a common denominator. This gives a(1 − 2x) + b(1 − x) 1 . (1 − x)(1 − 2x) (1 − x)(1 − 2x) so 1 a(1 − 2x) + b(1 − x). This must be true for all values of x. If x 1, then the equation becomes 1 −a so a −1. When x 21 we get 1 b/2 so b 2. This tells us that we can decompose the fraction like this: 1 −1 2 + . (1 − x)(1 − 2x) 1 − x 1 − 2x This completes the partial fraction decomposition. Notice that these two fractions are generating functions we know. In fact, we should be able to expand each of them. −1 −1−x−x 2 −x 3 −x 4 −· · · which generates −1, −1, −1, −1, −1, . . . . 1−x 2 2+4x+8x 2 +16x 3 +32x 4 +· · · which generates 2, 4, 8, 16, 32, . . . . 1 − 2x We can give a closed formula for the nth term of each of these sequences. The ﬁrst is just a n −1. The second is a n 2n+1 . The sequence we are interested in is just the sum of these, so the solution to the recurrence relation is a n 2n+1 − 1. 304 5. Additional Topics We can now add generating functions to our list of methods for solving recurrence relations. Exercises 1. Find the generating function for each of the following sequences by relating them back to a sequence with known generating function. (a) 4, 4, 4, 4, 4, . . .. (b) 2, 4, 6, 8, 10, . . .. (c) 0, 0, 0, 2, 4, 6, 8, 10, . . .. (d) 1, 5, 25, 125, . . .. (e) 1, −3, 9, −27, 81, . . .. (f) 1, 0, 5, 0, 25, 0, 125, 0, . . .. (g) 0, 1, 0, 0, 2, 0, 0, 3, 0, 0, 4, 0, 0, 5, . . .. 2. Find the sequence generated by the following generating functions: 4x (a) . 1−x 1 (b) . 1 − 4x x (c) . 1+x 3x (d) . (1 + x)2 (e) 3. 1 + x + x2 (Hint: multiplication). (1 − x)2 Show how you can get the generating function for the triangular numbers in three diﬀerent ways: (a) Take two derivatives of the generating function for 1, 1, 1, 1, 1, . . . (b) Use diﬀerencing. (c) Multiply two known generating functions. 4. Use diﬀerencing to ﬁnd the generating function for 4, 5, 7, 10, 14, 19, 25, . . .. 5. Find a generating function for the sequence with recurrence relation a n 3a n−1 − a n−2 with initial terms a 0 1 and a 1 5. 6. Use the recurrence relation for the Fibonacci numbers to ﬁnd the generating function for the Fibonacci sequence. 5.1. Generating Functions 305 7. Use multiplication to ﬁnd the generating function for the sequence of partial sums of Fibonacci numbers, S0 , S1 , S2 , . . . where S0 F0 , S1 F0 + F1 , S2 F0 + F1 + F2 , S3 F0 + F1 + F2 + F3 and so on. 8. Find the generating function for the sequence with closed formula a n 2(5n ) + 7(−3)n . 9. Find a closed formula for the nth term of the sequence with generating 3x 1 function + . 1 − 4x 1 − x 10. Find a 7 for the sequence with generating function 11. Explain how we know that 1, 2, 3, 4, . . .. 2 x · . 2 (1 − x) 1 − x − x 2 1 is the generating function for (1 − x)2 12. Starting with the generating function for 1, 2, 3, 4, . . ., ﬁnd a generating function for each of the following sequences. (a) 1, 0, 2, 0, 3, 0, 4, . . .. (b) 1, −2, 3, −4, 5, −6, . . .. (c) 0, 3, 6, 9, 12, 15, 18, . . .. (d) 0, 3, 9, 18, 30, 45, 63, . . .. (Hint: relate this sequence to the previous one.) 1 1 − x − x2 (because it does). Use this fact to ﬁnd the sequence generated by each of the following generating functions. 13. You may assume that 1, 1, 2, 3, 5, 8, . . . has generating function (a) x2 . 1−x−x 2 (b) 1 . 1−x 2 −x 4 (c) 1 . 1−3x−9x 2 (d) 1 . (1−x−x 2 )(1−x) 14. Find the generating function for the sequence 1, −2, 4, −8, 16, . . .. 15. Find the generating function for the sequence 1, 1, 1, 2, 3, 4, 5, 6, . . .. 16. Suppose A is the generating function for the sequence 3, 5, 9, 15, 23, 33, . . .. (a) Find a generating function (in terms of A) for the sequence of diﬀerences between terms. (b) Write the sequence of diﬀerences between terms and ﬁnd a generating function for it (without referencing A). 306 5. Additional Topics (c) Use your answers to parts (a) and (b) to ﬁnd the generating function for the original sequence. 5.2. Introduction to Number Theory 5.2 307 Introduction to Number Theory We have used the natural numbers to solve problems. This was the right set of numbers to work with in discrete mathematics because we always dealt with a whole number of things. The natural numbers have been a tool. Let’s take a moment now to inspect that tool. What mathematical discoveries can we make about the natural numbers themselves? This is the main question of number theory: a huge, ancient, complex, and above all, beautiful branch of mathematics. Historically, number theory was known as the Queen of Mathematics and was very much a branch of pure mathematics, studied for its own sake instead of as a means to understanding real world applications. This has changed in recent years however, as applications of number theory have been unearthed. Probably the most well known example of this is RSA cryptography, one of the methods used in encrypt data on the internet. It is number theory that makes this possible. What sorts of questions belong to the realm of number theory? Here is a motivating example. Recall in our study of induction, we asked: Which amounts of postage can be made exactly using just 5-cent and 8-cent stamps? We were able to prove that any amount greater than 27 cents could be made. You might wonder what would happen if we changed the denomination of the stamps. What if we instead had 4- and 9-cent stamps? Would there be some amount after which all amounts would be possible? Well, again, we could replace two 4-cent stamps with a 9-cent stamp, or three 9-cent stamps with seven 4-cent stamps. In each case we can create one more cent of postage. Using this as the inductive case would allow us to prove that any amount of postage greater than 23 cents can be made. What if we had 2-cent and 4-cent stamps. Here it looks less promising. If we take some number of 2-cent stamps and some number of 4-cent stamps, what can we say about the total? Could it ever be odd? Doesn’t look like it. Why does 5 and 8 work, 4 and 9 work, but 2 and 4 not work? What is it about these numbers? If I gave you a pair of numbers, could you tell me right away if they would work or not? We will answer these questions, and more, after ﬁrst investigating some simpler properties of numbers themselves. Divisibility It is easy to add and multiply natural numbers. If we extend our focus to all integers, then subtraction is also easy (we need the negative numbers 308 5. Additional Topics so we can subtract any number from any other number, even larger from smaller). Division is the ﬁrst operation that presents a challenge. If we wanted to extend our set of numbers so any division would be possible (maybe excluding division by 0) we would need to look at the rational numbers (the set of all numbers which can be written as fractions). This would be going too far, so we will refuse this option. In fact, it is a good thing that not every number can be divided by other numbers. This helps us understand the structure of the natural numbers and opens the door to many interesting questions and applications. If given numbers a and b, it is possible that a ÷ b gives a whole number. In this case, we say that b divides a, in symbols, we write b | a. If this holds, then b is a divisor or factor of a, and a is a multiple of b. In other words, if b | a, then a bk for some integer k (this is saying a is some multiple of b). The Divisibility Relation. Given integers m and n, we say “m divides n” and write m|n provided n ÷ m is an integer. Thus the following assertions mean the same thing: 1. m | n 2. n mk for some integer k 3. m is a factor (or divisor) of n 4. n is a multiple of m. Notice that m | n is a statement. It is either true or false. On the other hand, n ÷ m or n/m is some number. If we want to claim that n/m is not an integer, so m does not divide n, then we can write m ∤ n. Example 5.2.1 Decide whether each of the statements below are true or false. 1. 4 | 20 4. 5 | 0 7. −3 | 12 2. 20 | 4 5. 7 | 7 8. 8 | 12 3. 0 | 5 6. 1 | 37 9. 1642 | 136299 Solution. 5.2. Introduction to Number Theory 309 1. True. 4 “goes into” 20 ﬁve times without remainder. In other words, 20 ÷ 4 5, an integer. We could also justify this by saying that 20 is a multiple of 4: 20 4 · 5. 2. False. While 20 is a multiple of 4, it is false that 4 is a multiple of 20. 3. False. 5 ÷ 0 is not even deﬁned, let alone an integer. 4. True. In fact, x | 0 is true for all x. This is because 0 is a multiple of every number: 0 x · 0. 5. True. In fact, x | x is true for all x. 6. True. 1 divides every number (other than 0). 7. True. Negative numbers work just ﬁne for the divisibility relation. Here 12 −3 · 4. It is also true that 3 | −12 and that −3 | −12. 8. False. Both 8 and 12 are divisible by 4, but this does not mean that 12 is divisible by 8. 9. False. See below. This last example raises a question: how might one decide whether m | n? Of course, if you had a trusted calculator, you could ask it for the value of n ÷ m. If it spits out anything other than an integer, you know m ∤ n. This seems a little like cheating though: we don’t have division, so should we really use division to check divisibility? While we don’t really know how to divide, we do know how to multiply. We might try multiplying m by larger and larger numbers until we get close to n. How close? Well, we want to be sure that if we multiply m by the next larger integer, we go over n. For example, let’s try this to decide whether 1642 | 136299. Start ﬁnding multiples of 1642: 1642 · 2 3284 1642 · 3 4926 1642 · 4 6568 ···. All of these are well less than 136299. I suppose we can jump ahead a bit: 1642 · 50 82100 1642 · 80 131360 1642 · 85 139570. Ah, so we need to look somewhere between 80 and 85. Try 83: 1642 · 83 136286. Is this the best we can do? How far are we from our desired 136299? If we subtract, we get 136299 − 136286 13. So we know we cannot go up 310 5. Additional Topics to 84, that will be too much. In other words, we have found that 136299 83 · 1642 + 13. Since 13 < 1642, we can now safely say that 1642 ∤ 136299. It turns out that the process we went through above can be repeated for any pair of numbers. We can always write the number a as some multiple of the number b plus some remainder. We know this because we know about division with remainder from elementary school. This is just a way of saying it using multiplication. Due to the procedural nature that can be used to ﬁnd the remainder, this fact is usually called the division algorithm: The Division Algorithm. Given any two integers a and b, we can always ﬁnd an integer q such that a qb + r where r is an integer satisfying 0 ≤ r < |b| The idea is that we can always take a large enough multiple of b so that the remainder r is as small as possible. We do allow the possibility of r 0, in which case we have b | a. Remainder Classes The division algorithm tells us that there are only b possible remainders when dividing by b. If we ﬁx this divisor, we can group integers by the remainder. Each group is called a remainder class modulo b (or sometimes residue class). Example 5.2.2 Describe the remainder classes modulo 5. Solution. We want to classify numbers by what their remainder would be when divided by 5. From the division algorithm, we know there will be exactly 5 remainder classes, because there are only 5 choices for what r could be (0 ≤ r < 5). First consider r 0. Here we are looking for all the numbers divisible by 5 since a 5q + 0. In other words, the multiples of 5. We get the inﬁnite set {. . . , −15, −10, −5, 0, 5, 10, 15, 20, . . .}. 5.2. Introduction to Number Theory 311 Notice we also include negative integers. Next consider r 1. Which integers, when divided by 5, have remainder 1? Well, certainly 1, does, as does 6, and 11. Negatives? Here we must be careful: −6 does NOT have remainder 1. We can write −6 −2 · 5 + 4 or −6 −1 · 5 − 1, but only one of these is a “correct” instance of the division algorithm: r 4 since we need r to be non-negative. So in fact, to get r 1, we would have −4, or −9, etc. Thus we get the remainder class {. . . , −14, −9, −4, 1, 6, 11, 16, 21, . . .}. There are three more to go. The remainder classes for 2, 3, and 4 are, respectively {. . . , −13, −8, −3, 2, 7, 12, 17, 22, . . .} {. . . , −12, −7, −2, 3, 8, 13, 18, 23, . . .} {. . . , −11, −6, −1, 4, 9, 14, 19, 24, . . .}. Note that in the example above, every integer is in exactly one remainder class. The technical way to say this is that the remainder classes modulo b form a partition of the integers.1 The most important fact about partitions, is that it is possible to deﬁne an equivalence relation from a partition: this is a relationship between pairs of numbers which acts in all the important ways like the “equals” relationship.2 All fun technical language aside, the idea is really simple. If two numbers belong to the same remainder class, then in some way, they are the same. That is, they are the same up to division by b. In the case where b 5 above, the numbers 8 and 23, while not the same number, are the same when it comes to dividing by 5, because both have remainder 3. It matters what the divisor is: 8 and 23 are the same up to division by 5, but not up to division by 7, since 8 has remainder of 1 when divided by 7 while 23 has a remainder of 2. With all this in mind, let’s introduce some notation. We want to say that 8 and 23 are basically the same, even though they are not equal. It would be wrong to say 8 23. Instead, we write 8 ≡ 23. But this is not always true. It works if we are thinking division by 5, so we need to denote that somehow. What we will actually write is this: 8 ≡ 23 (mod 5) 1It is possible to develop a mathematical theory of partitions, prove statements about all partitions in general and then apply those observations to our case here. 2Again, there is a mathematical theory of equivalence relations which applies in many more instances than the one we look at here. 312 5. Additional Topics which is read, “8 is congruent to 23 modulo 5” (or just “mod 5”). Of course then we could observe that 8 . 23 (mod 7). Congruence Modulo n. We say a is congruent to b modulo n, and write, a≡b (mod n) provided a and b have the same remainder when divided by n. In other words, provided a and b belong to the same remainder class modulo n. Many books deﬁne congruence modulo n slightly diﬀerently. They say that a ≡ b (mod n) if and only if n | a − b. In other words, two numbers are congruent modulo n, if their diﬀerence is a multiple of n. So which deﬁnition is correct? Turns out, it doesn’t matter: they are equivalent. To see why, consider two numbers a and b which are congruent modulo n. Then a and b have the same remainder when divided by n. We have a q1 n + r b q 2 n + r. Here the two r’s really are the same. Consider what we get when we take the diﬀerence of a and b: a − b q1 n + r − (q2 n + r) q1 n − q2 n (q1 − q2 )n. So a − b is a multiple of n, or equivalently, n | a − b. On the other hand, if we assume ﬁrst that n | a − b, so a − b kn, then consider what happens if we divide each term by n. Dividing a by n will leave some remainder, as will dividing b by n. However, dividing kn by n will leave 0 remainder. So the remainders on the left-hand side must cancel out. That is, the remainders must be the same. Thus we have: Congruence and Divisibility. For any integers a, b, and n, we have a≡b (mod n) if and only if n | a − b. It will also be useful to switch back and forth between congruences and regular equations. The above fact helps with this. We know that a ≡ b (mod n) if and only if n | a − b, if and only if a − b kn for some 5.2. Introduction to Number Theory 313 integer k. Rearranging that equation, we get a b + kn. In other words, if a and b are congruent modulo n, then a is b more than some multiple of n. This conforms with our earlier observation that all the numbers in a particular remainder class are the same amount larger than the multiples of n. Congruence and Equality. For any integers a, b, and n, we have a≡b (mod n) if and only if a b+kn for some integer k. Properties of Congruence We said earlier that congruence modulo n behaves, in many important ways, the same way equality does. Speciﬁcally, we could prove that congruence modulo n is an equivalence relation, which would require checking the following three facts: Congruence Modulo n is an Equivalence Relation. Given any integers a, b, and c, and any positive integer n, the following hold: 1. a ≡ a (mod n). 2. If a ≡ b (mod n) then b ≡ a (mod n). 3. If a ≡ b (mod n) and b ≡ c (mod n), then a ≡ c (mod n). In other words, congruence modulo n is reﬂexive, symmetric, and transitive, so is an equivalence relation. You should take a minute to convince yourself that each of the properties above actually hold of congruence. Try explaining each using both the remainder and divisibility deﬁnitions. Next, consider how congruence behaves when doing basic arithmetic. We already know that if you subtract two congruent numbers, the result will be congruent to 0 (be a multiple of n). What if we add something congruent to 1 to something congruent to 2? Will we get something congruent to 3? Congruence and Arithmetic. Suppose a ≡ b (mod n) and c ≡ d (mod n). Then the following hold: 314 5. Additional Topics 1. a + c ≡ b + d (mod n). 2. a − c ≡ b − d (mod n). 3. ac ≡ bd (mod n). The above facts might be written a little strangely, but the idea is simple. If we have a true congruence, and we add the same thing to both sides, the result is still a true congruence. This sounds like we are saying: If a ≡ b (mod n) then a + c ≡ b + c (mod n). Of course this is true as well, it is the special case where c d. But what we have works in more generality. Think of congruence as being “basically equal.” If we have two numbers which are basically equal, and we add basically the same thing to both sides, the result will be basically equal. This seems reasonable. Is it really true? Let’s prove the ﬁrst fact: Proof. Suppose a ≡ b (mod n) and c ≡ d (mod n). That means a b + kn and c d + jn for integers k and j. Add these equations: a + c b + d + kn + jn. But kn + jn (k + j)n, which is just a multiple of n. So a + c b + d + (j + k)n, or in other words, a + c ≡ b + d (mod n) qed The other two facts can be proved in a similar way. One of the important consequences of these facts about congruences, is that we can basically replace any number in a congruence with any other number it is congruent to. Here are some examples to see how (and why) that works: Example 5.2.3 Find the remainder of 3491 divided by 9. Solution. We could do long division, but there is another way. We want to ﬁnd x such that x ≡ 3491 (mod 9). Now 3491 3000 + 400 + 90 + 1. Of course 90 ≡ 0 (mod 9), so we can replace the 90 in the sum with 0. Why is this okay? We are actually subtracting the “same” thing from both sides: x ≡ 3000 + 400 + 90 + 1 (mod 9) − 0 ≡ 90 (mod 9) x ≡ 3000 + 400 + 0 + 1 (mod 9). 5.2. Introduction to Number Theory 315 Next, note that 400 4 · 100, and 100 ≡ 1 (mod 9) (since 9 | 99). So we can in fact replace the 400 with simply a 4. Again, we are appealing to our claim that we can replace congruent elements, but we are really appealing to property 3 about the arithmetic of congruence: we know 100 ≡ 1 (mod 9), so if we multiply both sides by 4, we get 400 ≡ 4 (mod 9). Similarly, we can replace 3000 with 3, since 1000 1 + 999 ≡ 1 (mod 9). So our original congruence becomes x ≡ 3 + 4 + 0 + 1 (mod 9) x≡8 (mod 9). Therefore 3491 divided by 9 has remainder 8. The above example should convince you that the well known divisibility test for 9 is true: the sum of the digits of a number is divisible by 9 if and only if the original number is divisible by 9. In fact, we now know something more: any number is congruent to the sum of its digits, modulo 9.3 Let’s try another: Example 5.2.4 Find the remainder when 3123 is divided by 7. Solution. Of course, we are working with congruence because we want to ﬁnd the smallest positive x such that x ≡ 3123 (mod 7). Now ﬁrst write 3123 (33 )41 . We have: 3123 2741 ≡ 641 (mod 7), since 27 ≡ 6 (mod 7). Notice further that 62 36 is congruent to 1 modulo 7. Thus we can simplify further: 641 6 · (62 )20 ≡ 6 · 120 (mod 7). But 120 1, so we are done: 3123 ≡ 6 (mod 7). 3This works for 3 as well, but deﬁnitely not for any modulus in general. 316 5. Additional Topics In the above example, we are using the fact that if a ≡ b (mod n), then a p ≡ b p (mod n). This is just applying property 3 a bunch of times. So far we have seen how to add, subtract and multiply with congruences. What about division? There is a reason we have waited to discuss it. It turns out that we cannot simply divide. In other words, even if ad ≡ bd (mod n), we do not know that a ≡ b (mod n). Consider, for example: 18 ≡ 42 (mod 8). This is true. Now 18 and 42 are both divisible by 6. However, 3 . 7 (mod 8). While this doesn’t work, note that 3 ≡ 7 (mod 4). We cannot divide 8 by 6, but we can divide 8 by the greatest common factor of 8 and 6. Will this always happen? Suppose ad ≡ bd (mod n). In other words, we have ad bd + kn for some integer k. Of course ad is divisible by d, as is bd. So kn must also be divisible by d. Now if n and d have no common factors (other than 1), then we must have d | k. But in general, if we try to divide kn by d, we don’t know that we will get an integer multiple of n. Some of the n might get divided as well. To be safe, let’s divide as much of n as we can. Take the largest factor of both d and n, and cancel that out from n. The rest of the factors of d will come from k, no problem. We will call the largest factor of both d and n the gcd(d, n), for greatest common divisor. In our example above, gcd(6, 8) 2 since the greatest divisor common to 6 and 8 is 2. Congruence and Division. n Suppose ad ≡ bd (mod n). Then a ≡ b (mod gcd(d,n) ). If d and n have no common factors then gcd(d, n) 1, so a ≡ b (mod n). Example 5.2.5 Simplify the following congruences using division: (a) 24 ≡ 39 (mod 5) and (b) 24 ≡ 39 (mod 15). Solution. (a) Both 24 and 39 are divisible by 3, and 3 and 5 have no common factors, so we get 8 ≡ 13 (mod 5). (b) Again, we can divide by 3. However, doing so blindly gives us 8 ≡ 13 (mod 15) which is no longer true. Instead, we must also divide the modulus 15 by the greatest common factor of 3 and 15, 5.2. Introduction to Number Theory 317 which is 3. Again we get 8 ≡ 13 (mod 5). Solving Congruences Now that we have some algebraic rules to govern congruence relations, we can attempt to solve for an unknown in a congruence. For example, is there a value of x that satisﬁes, 3x + 2 ≡ 4 (mod 5), and if so, what is it? In this example, since the modulus is small, we could simply try every possible value for x. There are really only 5 to consider, since any integer that satisﬁed the congruence could be replaced with any other integer it was congruent to modulo 5. Here, when x 4 we get 3x + 2 14 which is indeed congruent to 4 modulo 5. This means that x 9 and x 14 and x 19 and so on will each also be a solution because as we saw above, replacing any number in a congruence with a congruent number does not change the truth of the congruence. So in this example, simply compute 3x+2 for values of x ∈ {0, 1, 2, 3, 4}. This gives 2, 5, 8, 11, and 14 respectively, for which only 14 is congruent to 4. Let’s also see how you could solve this using our rules for the algebra of congruences. Such an approach would be much simpler than the trial and error tactic if the modulus was larger. First, we know we can subtract 2 from both sides: 3x ≡ 2 (mod 5). Then to divide both sides by 3, we ﬁrst add 0 to both sides. Of course, on the right-hand side, we want that 0 to be a 10 (yes, 10 really is 0 since they are congruent modulo 5). This gives, 3x ≡ 12 (mod 5). Now divide both sides by 3. Since gcd(3, 5) 1, we do not need to change the modulus: x ≡ 4 (mod 5). Notice that this in fact gives the general solution: not only can x 4, but x can be any number which is congruent to 4. We can leave it like this, or write “x 4 + 5k for any integer k.” 318 5. Additional Topics Example 5.2.6 Solve the following congruences for x. 1. 7x ≡ 12 (mod 13). 2. 84x − 38 ≡ 79 (mod 15). 3. 20x ≡ 23 (mod 14). Solution. 1. All we need to do here is divide both sides by 7. We add 13 to the right-hand side repeatedly until we get a multiple of 7 (adding 13 is the same as adding 0, so this is legal). We get 25, 38, 51, 64, 77 – got it. So we have: 7x ≡ 12 (mod 13) 7x ≡ 77 (mod 13) x ≡ 11 (mod 13). 2. Here, since we have numbers larger than the modulus, we can reduce them prior to applying any algebra. We have 84 ≡ 9, 38 ≡ 8 and 79 ≡ 4. Thus, 84x − 38 ≡ 79 (mod 15) 9x − 8 ≡ 4 (mod 15) 9x ≡ 12 (mod 15) 9x ≡ 72 (mod 15). We got the 72 by adding 0 ≡ 60 (mod 15) to both sides of the congruence. Now divide both sides by 9. However, since gcd(9, 15) 3, we must divide the modulus by 3 as well: x≡8 (mod 5). So the solutions are those values which are congruent to 8, or equivalently 3, modulo 5. This means that in some sense there are 3 solutions modulo 15: 3, 8, and 13. We can write the solution: x≡3 (mod 15); x ≡ 8 (mod 15); x ≡ 13 (mod 15). 3. First, reduce modulo 14: 20x ≡ 23 (mod 14) 5.2. Introduction to Number Theory 319 6x ≡ 9 (mod 14). We could now divide both sides by 3, or try to increase 9 by a multiple of 14 to get a multiple of 6. If we divide by 3, we get, 2x ≡ 3 (mod 14). Now try adding multiples of 14 to 3, in hopes of getting a number we can divide by 2. This will not work! Every time we add 14 to the right side, the result will still be odd. We will never get an even number, so we will never be able to divide by 2. Thus there are no solutions to the congruence. The last congruence above illustrates the way in which congruences might not have solutions. We could have seen this immediately in fact. Look at the original congruence: 20x ≡ 23 (mod 14). If we write this as an equation, we get 20x 23 + 14k, or equivalently 20x − 14k 23. We can easily see there will be no solution to this equation in integers. The left-hand side will always be even, but the right-hand side is odd. A similar problem would occur if the right-hand side was divisible by any number the left-hand side was not. So in general, given the congruence ax ≡ b (mod n), if a and n are divisible by a number which b is not divisible by, then there will be no solutions. In fact, we really only need to check one divisor of a and n: the greatest common divisor. Thus, a more compact way to say this is: Congruences with no solutions. If gcd(a, n) ∤ b, then ax ≡ b (mod n) has no solutions. Solving Linear Diophantine Equations Discrete math deals with whole numbers of things. So when we want to solve equations, we usually are looking for integer solutions. Equations which are intended to only have integer solutions were ﬁrst studied by in the third century by the Greek mathematician Diophantus of Alexandria, 320 5. Additional Topics and as such are called Diophantine equations. Probably the most famous example of a Diophantine equation is a 2 + b 2 c 2 . The integer solutions to this equation are called Pythagorean triples. In general, solving Diophantine equations is hard (in fact, there is provably no general algorithm for deciding whether a Diophantine equation has a solution, a result known as Matiyasevich’s Theorem). We will restrict our focus to linear Diophantine equations, which are considerably easier to work with. Diophantine Equations. An equation in two or more variables is called a Diophantine equation if only integers solutions are of interest. A linear Diophantine equation takes the form a 1 x 1 + a 2 x x + · · · + a n x n b for constants a 1 , . . . , a n , b. A solution to a Diophantine equation is a solution to the equation consisting only of integers. We have the tools we need to solve linear Diophantine equations. We will consider, as a main example, the equation 51x + 87y 123. The general strategy will be to convert the equation to a congruence, then solve that congruence.4 Let’s work this particular example to see how this might go. First, check if perhaps there are no solutions because a divisor of 51 and 87 is not a divisor of 123. Really, we just need to check whether gcd(51, 87) | 123. This greatest common divisor is 3, and yes 3 | 123. At this point, we might as well factor out this greatest common divisor. So instead, we will solve: 17x + 29y 41. Now observe that if there are going to be solutions, then for those values of x and y, the two sides of the equation must have the same remainder as each other, no matter what we divide by. In particular, if we divide both sides by 17, we must get the same remainder. Thus we can safely write 17x + 29y ≡ 41 (mod 17). We choose 17 because 17x will have remainder 0. This will allow us to reduce the congruence to just one variable. We could have also moved to a congruence modulo 29, although there is usually a good reason to select 4This is certainly not the only way to proceed. A more common technique would be to apply the Euclidean algorithm. Our way can be a little faster, and is presented here primarily for variety. 5.2. Introduction to Number Theory 321 the smaller choice, as this will allow us to reduce the other coeﬃcient. In our case, we reduce the congruence as follows: 17x + 29y ≡ 41 (mod 17) 0x + 12y ≡ 7 (mod 17) 12y ≡ 24 (mod 17) y≡2 (mod 17). Now at this point we know y 2 + 17k will work for any integer k. If we haven’t made a mistake, we should be able to plug this back into our original Diophantine equation to ﬁnd x: 17x + 29(2 + 17k) 41 17x −17 − 29 · 17k x −1 − 29k. We have now found all solutions to the Diophantine equation. For each k, x −1 − 29k and y 2 + 17k will satisfy the equation. We could check this for a few cases. If k 0, the solution is (−1, 2), and yes, −17 + 2 · 29 41. If k 3, the solution is (−88, 53). If k −2, we get (57, −32). To summarize this process, to solve ax + b y c, we, 1. Divide both sides of the equation by gcd(a, b) (if this does not leave the right-hand side as an integer, there are no solutions). Let’s assume that ax + b y c has already been reduced in this way. 2. Pick the smaller of a and b (here, assume it is b), and convert to a congruence modulo b: ax + b y ≡ c (mod b). This will reduce to a congruence with one variable, x: ax ≡ c (mod b). 3. Solve the congruence as we did in the previous section. Write your solution as an equation, such as, x n + kb. 4. Plug this into the original Diophantine equation, and solve for y. 5. If we want to know solutions in a particular range (for example, 0 ≤ x, y ≤ 20), pick diﬀerent values of k until you have all required solutions. Here is another example: 322 5. Additional Topics Example 5.2.7 How can you make $6.37 using just 5-cent and 8-cent stamps? What is the smallest and largest number of stamps you could use? Solution. First, we need a Diophantine equation. We will work in numbers of cents. Let x be the number of 5-cent stamps, and y be the number of 8-cent stamps. We have: 5x + 8y 637. Convert to a congruence and solve: 8y ≡ 367 (mod 5) 3y ≡ 2 (mod 5) 3y ≡ 12 (mod 5) y ≡ 4 (mod 5). Thus y 4 + 5k. Then 5x + 8(4 + 5k) 637, so x 121 − 8k. This says that one way to make $6.37 is to take 121 of the 5-cent stamps and 4 of the 8-cent stamps. To ﬁnd the smallest and largest number of stamps, try diﬀerent values of k. k (x, y) Stamps -1 0 1 2 .. . (129, -1) (121, 4) (113, 9) (105, 13) .. . not possible 125 122 119 .. . This is no surprise. Having the most stamps means we have as many 5-cent stamps as possible, and to get the smallest number of stamps would require have the least number of 5-cent stamps. To minimize the number of 5-cent stamps, we want to pick k so that 121 − 8k is as small as possible (but still positive). When k 15, we have x 1 and y 79. Therefore, to make $6.37, you can us as few as 80 stamps (1 5-cent stamp and 79 8-cent stamps) or as many as 125 stamps (121 5-cent stamps and 4 8-cent stamps). Using this method, as long as you can solve linear congruences in one variable, you can solve linear Diophantine equations of two variables. There are times though that solving the linear congruence is a lot of work. 5.2. Introduction to Number Theory 323 For example, suppose you need to solve, 13x ≡ 6 (mod 51). You could keep adding 51 to the right side until you get a multiple of 13: You would get 57, 108, 159, 210, 261, 312, and 312 is the ﬁrst of these that is divisible by 13. This works, but is really too much work. Instead we could convert back to a Diophantine equation: 13x 6 + 51k. Now solve this like we have in this section. Write it as a congruence modulo 13: 0 ≡ 6 + 51k (mod 13) −12k ≡ 6 2k ≡ −1 (mod 13) (mod 13) 2k ≡ 12 (mod 13) k≡6 (mod 13). so k 6 + 13j. Now go back and ﬁgure out x: 13x 6 + 51(6 + 13j) x 24 + 51j. Of course you could do this switching back and forth between congruences and Diophantine equations as many times as you like. If you only used this technique, you would essentially replicate the Euclidean algorithm, a more standard way to solve Diophantine equations. Exercises 1. Suppose a, b, and c are integers. Prove that if a | b, then a | bc. 2. Suppose a, b, and c are integers. Prove that if a | b and a | c then a | b + c and a | b − c. 3. Write out the remainder classes for n 4. 4. What is the largest n such that 16 and 25 are in the same remainder class modulo n? Write out the remainder class they both belong to and give an example of a number more than 100 in that class. 5. Let a, b, c, and n be integers. Prove that if a ≡ b (mod n) and c ≡ d (mod n), then a − c ≡ b − d (mod n). 6. Find the remainder of 3456 when divided by (a) 2. (b) 5. (c) 7. 7. Repeat the previous exercise, this time dividing 22019 . (d) 9. 324 8. 5. Additional Topics Determine which of the following congruences have solutions, and ﬁnd any solutions (between 0 and the modulus) by trial and error. (a) 4x ≡ 5 (mod 6). (b) 6x ≡ 3 (mod 9). (c) x 2 ≡ 2 (mod 4). 9. Determine which of the following congruences have solutions, and ﬁnd any solutions (between 0 and the modulus) by trial and error. (a) 4x ≡ 5 (mod 7). (b) 6x ≡ 4 (mod 9). (c) x 2 ≡ 2 (mod 7). 10. Solve the following congruence 5x+8 ≡ 11 (mod 22). That is, describe the general solution. 11. Solve the congruence: 6x ≡ 4 (mod 10). 12. Solve the congruence: 4x ≡ 24 (mod 30). 13. Solve the congruence: 341x ≡ 2941 (mod 9). 14. I’m thinking of a number. If you multiply my number by 7, add 5, and divide the result by 11, you will be left with a remainder of 2. What remainder would you get if you divided my original number by 11? 15. Solve the following linear Diophantine equation, using modular arithmetic (describe the general solutions). 6x + 10y 32. 16. Solve the following linear Diophantine equation, using modular arithmetic (describe the general solutions). 17x + 8y 31. 17. Solve the following linear Diophantine equation, using modular arithmetic (describe the general solutions). 35x + 47y 1. 18. You have a 13 oz. bottle and a 20 oz. bottle, with which you wish to measure exactly 2 oz. However, you have a limited supply of water. If any water enters either bottle and then gets dumped out, it is gone forever. What is the least amount of water you can start with and still complete the task? Appendix A Selected Hints 0.2 Exercises 0.2.18. Try an example. What if P(x) was the predicate, “x is prime”? What if it was “if x is divisible by 4, then it is even”? Of course examples are not enough to prove something in general, but that is entirely the point of this question. 0.2.19. First ﬁgure out what each statement is saying. For part (c), you don’t need to assume the domain is an inﬁnite set. 0.3 Exercises 0.3.7. You should be able to write all of them out. Don’t forget A and B, which are also candidates for C. 0.3.14. It might help to think about what the union A2 ∪ A3 is ﬁrst. Then think about what numbers are not in that union. What will happen when you also include A5 ? 0.3.17. We are looking for a set containing 16 sets. 0.3.18. Write these out, or at least start to and look for a pattern. 0.3.29. It looks like you should be able to deﬁne the set A like this. But consider the two possible values for |A|. 0.4 Exercises 0.4.20. 1 2 3 and g Work with some examples. What if f a a b a b c ? 5 6 7 0.4.25. To ﬁnd the recurrence relation, consider how many new handshakes occur when person n + 1 enters the room. 0.4.29. One of these is not always true. Try some examples! 1.1 Exercises 1.1.9. To ﬁnd out how many numbers are divisible by 6 and 7, for example, take 500/42 and round down. 1.1.11. For part (a) you could use the formula for PIE, but for part (b) you might be better oﬀ drawing a Venn diagram. 325 326 A. Selected Hints 1.1.13. You could consider cases. For example, any number of the form ODD-ODD-EVEN will have an even sum. Alternatively, how many three digit numbers have the sum of their digits even if the ﬁrst two digits are 54? What if the ﬁrst two digits are 19? 1.1.14. For a simpler example, there are 4 divisors of 6 2 · 3. They are 1 20 · 30 , 2 21 · 30 , 3 20 · 31 and 6 21 · 31 . 1.2 Exercises 1.2.5. Pennies are sort of like 0’s and nickels are sort of like 1’s. 1.2.7. Break the question into ﬁve cases. 1.3 Exercises 1.3.4. Which question should have the larger answer? One of these is a combination, the other is a permutation. 1.3.6. If you pick any three points, you can get a triangle, unless those three points are all on the x-axis or on the y-axis. There are other ways to start this as well, and any correct method should give the same answer. 1.3.8. We just need a string of 7 letters: 4 of one type, 3 of the other. 1.3.11. There are 10 people seated around the table, but it does not matter where King Arthur sits, only who sits to his left, two seats to his left, and so on. So the answer is not 10!. 1.4 Exercises 1.4.3. There will be 185 triangles. But to ﬁnd them . . . (a) How many vertices of the triangle can be on the horizontal axis? (b) Will any three dots work as the vertices? 1.4.4. The answer is 120. 1.4.6. Try Exercise 1.4.5 1.4.7. What if you wanted a pair of co-maids-of-honor? 1.4.8. For the combinatorial proof: what if you don’t yet know how many bridesmaids you will have? 1.4.9. Count handshakes. 1.4.13. This one might remind you of Example 1.4.6 1.4.14. For the lattice paths, think about what sort of paths 2n would count. Not all the paths will end at the same point, but you could describe the set of end points as a line. Selected Hints 327 1.6 Exercises 1.6.4. Instead, count the solutions to y1 + y2 + y3 + y4 7 with 0 ≤ y i ≤ 3. Why is this equivalent? 1.6.13. This is a sneaky way to ask for the number of derangements on 5 elements. Do you see why? 1.7 Chapter Review 1.7.16. Stars and bars. 2.1 Exercises 2.1.11. You will want to write out the sequence, guess a closed formula, and then verify that you are correct. 2.1.12. Write out the sequence, guess a recursive deﬁnition, and verify that the closed formula is a solution to that recursive deﬁnition. 2.1.15. Try an example: when you draw the 4th line, it will cross three other lines, so will be divided into four segments, two of which are inﬁnite. Each segment will divide a previous region into two. 2.1.16. Consider three cases: the last digit is a 0, a 1, or a 2. Two of these should be easy to count, but strings ending in 0 cannot be proceeded by a 2, so require a little more work. 2.1.18. Think recursively, like you did in Pascal’s triangle. 2.1.19. There is only one way to tile a 2 × 1 board, and two ways to tile a 2 × 2 board (you can orient the dominoes in two ways). In general, consider the two ways the domino covering the top left corner could be oriented. 2.4 Exercises 2.4.3. Use telescoping or iteration. 2.5 Exercises 2.5.9. It is not possible to score exactly 11 points. Can you prove that you can score n points for any n ≥ 12? 2.5.11. Start with (k + 1)-gon and divide it up into a k-gon and a triangle. 2.5.15. For the inductive step, you can assume you have a strictly increasing sequence up to a k where a k < 100. Now you just need to ﬁnd the next term a k+1 so that a k < a k+1 < 100. What should a k+1 be? 328 A. Selected Hints 2.5.17. For the inductive case, you will need to show that (k + 1)2 + (k + 1) is even. Factor this out and locate the part of it that is k 2 + k. What have you assumed about that quantity? 2.5.18. This is similar to Exercise 2.5.15, although there you were showing that a sequence had all its terms less than some value, and here you are showing that the sum is less than som value. But the partial sums forms a sequence, so this is actually very similar. 2.5.20. As with the previous question, we will want to subtract something from n in the inductive step. There we subtracted the largest power of 2 less than n. So what should you subtract here? Note, you will still need to take care here that the sum you get from the inductive hypothesis, together with the number you subtracted will be a sum of distinct Fibonacci numbers. In fact, you could prove that the Fibonacci numbers in the sum are non-consecutive! 2.5.21. We have already proved this without using induction, but looking at it inductively sheds light onto the problem (and is fun). The question you need to answer to complete the inductive step is, how many new handshakes take place when a person k + 1 enters the room. Why does adding this give you the correct formula? You will need to use strong induction. For the inductive case, try multiplying x k + x1k x + x1 and collect which terms together are integers. 2.5.22. 2.5.23. Here’s the idea: since every entry in Pascal’s Triangle is the sum of the two entries above it, we can get the k + 1st row by adding up all the pairs of entry from the kth row. But doing this uses each entry on the kth row twice. Thus each time we drop to the next row, we double the total. Of course, row 0 has sum 1 20 (the base case). Now try to make thisprecisewith a formal induction proof. You will use the fact that n n−1 n−1 for the inductive case. k k−1 + k 2.5.24. To see why this works, try it on a copy of Pascal’s triangle. We are adding up the entries along a diagonal, starting with the 1 on the left-hand side of the 4th row. Suppose we add up the ﬁrst 5 entries on this diagonal. The claim is that the sum is the entry below and to the left of the last of these 5 entries. Note that if this is true, and we instead add up the ﬁrst 6 entries, we will need to add the entry one spot to the right of the previous sum. But these two together give the entry below them, which is below and left of the last of the 6 entries on the diagonal. If you follow that, you can see what is going on. But it is not a great proof. A formal induction proof is needed. Selected Hints 329 2.5.26. You are allowed to assume the base case. For the inductive case, group all but the last function together as one sum of functions, then apply the usual sum of derivatives rule, and then the inductive hypothesis. 2.5.27. For the inductive step, we know by the product rule for two functions that ′ ( f1 f2 f3 · · · f k f k+1 )′ ( f1 f2 f3 · · · f k )′ f k+1 + ( f1 f2 f3 · · · f k ) f k+1 . Then use the inductive hypothesis on the ﬁrst summand, and distribute. 2.5.29. You will need three base cases. This is a very good hint actually, as it suggests that to prove P(n) is true, you would want to use the fact that P(n − 3) is true. So somehow you need to increase the number of squares by 3. 2.6 Chapter Review 2.6.14. (a) Hint: (n + 1)n+1 > (n + 1) · n n . (b) Hint: This should be similar to the other sum proofs. The last bit comes down to adding fractions. (c) Hint: Write 4k+1 − 1 4 · 4k − 4 + 3. (d) Hint: one 9-cent stamp is 1 more than two 4-cent stamps, and seven 4-cent stamps is 1 more than three 9-cent stamps. (e) Careful to actually use induction here. The base case: 22 4. The inductive case: assume (2n)2 is divisible by 4 and consider (2n + 2)2 (2n)2 + 4n + 4. This is divisible by 4 because 4n + 4 clearly is, and by our inductive hypothesis, so is (2n)2 . 2.6.15. This is a straight forward induction proof. Note you will need to simplify n(n+1) 2 2 + (n + 1)3 and get (n+1)(n+2) 2 2 . 2.6.16. There are two base cases P(0) and P(1). Then, for the inductive case, assume P(k) is true for all k < n. This allows you to assume a n−1 1 and a n−2 1. Apply the recurrence relation. 3.1 Exercises 3.1.4. Like above, only now you will need 8 rows instead of just 4. 3.1.5. You should write down three statements using the symbols P, Q, R, S. If Geoﬀ is a truth-teller, then all three statements would be true. If he was a liar, then all three statements would be false. But in 330 A. Selected Hints either case, we don’t yet know whether the four atomic statements are true or false, since he hasn’t said them by themselves. A truth table might help, although is probably not entirely necessary. 3.1.10. (a) There will be three rows in which the statement is false. (b) Consider the three rows that evaluate to false and say what the truth values of T, S, and P are there. (c) You are looking for a row in which P is true, and the whole statement is true. 3.1.11. Write down three statements, and then take the negation of each (since he is a liar). You should ﬁnd that Tommy ate one item and drank one item. (Q is for cucumber sandwiches.) 3.1.15. For the second part, you can inductively assume that from the ﬁrst n − 2 implications you can deduce P1 → Pn−1 . Then you are back in the case in part (a) again. 3.1.18. It might help to translate the statements into symbols and then use the formulaic rules to simplify negations (i.e., rules for quantiﬁers and De Morgan’s laws). After simplifying, you should get ∀x(¬E(x) ∧ ¬O(x)), for the ﬁrst one, for example. Then translate this back into English. 3.1.19. What do these concepts mean in terms of truth tables? 3.2 Exercises 3.2.5. One of the implications will be a direct proof, the other will be a proof by contrapositive. √ 3.2.6. This is really an exercise in modifying the proof that 2 is irrational. There you proved things were even; here they will be multiples of 3. 3.2.7. Part (a) should be a relatively easy direct proof. Look for a counterexample for part (b). 3.2.9. A proof by contradiction would be reasonable here, because then you get to assume that both a and b are odd. Deduce that c 2 is even, and therefore a multiple of 4 (why? and why is that a contradiction?). 3.2.11. Use a diﬀerent style of proof for each part. The last part should remind you of the pigeonhole principle, so mimicking that proof might be helpful. a 3.2.13. Note that if log(7) ba , then 7 10 b . Can any power of 7 be the same as a power of 10? Selected Hints 331 3.2.14. What if there were? Deduce that x must be odd, and continue towards a contradiction. 3.2.15. Prove the contrapositive by cases. There will be 4 cases to consider. 3.2.16. Your friend’s proof a proof, but of what? What implication follows from the given proof? Is that helpful? 3.2.18. Consider the set of numbers of friends that everyone has. If everyone had a diﬀerent number of friends, this set must contain 20 elements. Is that possible? Why not? 3.2.19. This feels like the pigeonhole principle, although a bit more complicated. At least, you could try to replicate the style of proof used by the pigeonhole principle. How would the episodes need to be spaced out so that no two of your sixty were exactly 4 apart? 4.1 Exercises 4.1.3. Both situations are possible. Go ﬁnd some examples. 4.1.6. The bipartite graph is a little tricky. You will deﬁnitely want a complete bipartite graph, but it could be K 5,5 or maybe K 1,9 , or . . . 4.1.7. The ﬁrst graph is bipartite, which can be seen by labeling it as follows. B A B A B Two of the remaining three are also bipartite. 4.1.8. C 4 is bipartite; C 5 is not. What about all the other values of n? 4.1.11. How many edges does K n have? One of the two graphs will not be connected (unless j 1). 4.1.12. You should be able to deduce everything directly from the deﬁnition. However, perhaps it would be helpful to know that the N stands for neighborhood. 4.1.13. Be careful to make sure the edges are not “directed.” In a graph, if a is adjacent to b, then b is adjacent to a. In the language of relations, we say that the edge relation is symmetric. 4.1.14. You might want to answer the questions for some speciﬁc values of n to get a feel for them, but your ﬁnal answers should be in terms of n. 332 A. Selected Hints 4.1.15. Try a small example ﬁrst: any graph with 8 vertices must have two vertices of the same degree. If not, what would the degree sequence be? 4.1.16. Use the handshake lemma 4.1.5. What would happen if all the vertices had degree 2? 4.2 Exercises 4.2.3. Careful: the graphs might not be connected. 4.2.4. Try Exercise 4.2.2. 4.2.5. Try a proof by contradiction and consider a spanning tree of the graph. 4.2.7. For part (b), trying some simple examples should give you the formula. Then you just need to prove it is correct. 4.2.8. Examining the proof of Proposition 4.2.1 gives you most of what you need, but make sure to just give the relevant parts, and take care to not use proof by contradiction. 4.2.9. You will need to remove a vertex of degree one, apply the inductive hypothesis to the result, and then say which set the degree one vertex to. 4.2.10. If e is the root, then b will have three children (a, c, and d), all of which will be siblings, and have b as their parent. a will not have any children. In general, how can you determine the number of children a vertex will have, if it is not a root? 4.2.14. There is an example with 7 edges. 4.2.15. The previous exercise will be helpful. 4.2.16. Note that such an edge, if removed, would disconnect the graph. We call graphs that have an edge like this 1-connected. 4.3 Exercises 4.3.3. What would Euler’s formula tell you? 4.3.5. You can use the handshake lemma to ﬁnd the number of edges, in terms of v, the number of vertices. 4.3.11. What is the length of the shortest cycle? (This quantity is usually called the girth of the graph.) 4.3.14. The girth of the graph is 4. 4.3.15. What has happened to the girth? Careful: we have a diﬀerent number of edges as well. Better check Euler’s formula. Selected Hints 333 4.4 Exercises 4.4.7. For (a), you will want the teams to be vertices and games to be edges. Which does it make sense to color? 4.4.10. The chromatic number is 4. Now prove this! Note that you cannot use the 4-color theorem, or Brooke’s theorem, or the clique number here. In fact, this graph, called the Grötzsch graph is the smallest graph with chromatic number 4 that does not contain any triangles. 4.4.13. You can color K 5 in such a way that every vertex is adjacent to exactly two blue edges and two red edges. However, there is a graph with only 5 edges that will result in a vertex incident to three edges of the same color no matter how they are colored. What is it, and how can you generalize? 4.4.14. The previous exercise is useful as a starting point. 4.5 Exercises 4.5.7. This is harder than the previous three questions. Think about which “side” of the graph the Hamilton path would need to be on every other step. 4.5.9. If you read oﬀ the names of the students in order, you would need to read each student’s name exactly once, and the last name would need to be of a student who was friends with the ﬁrst. What sort of a cycle is this? 4.5.10. Draw a graph with 6 vertices and 8 edges. What sort of path would be appropriate? 4.7 Chapter Review 4.7.23. You might want to give the proof in two parts. First prove by induction that the cycle C n has v e. Then consider what happens if the graph is more than just the cycle. 5.1 Exercises 5.1.10. You should “multiply” the two sequences. 5.2 Exercises 5.2.13. First reduce each number modulo 9, which can be done by adding up the digits of the numbers. 5.2.18. Solve the Diophantine equation 13x + 20y 2 (why?). Then consider which value of k (the parameter in the solution) is optimal. 334 A. Selected Hints Appendix B Selected Solutions 0.2 Exercises 0.2.1. (a) This is not a statement. It is an imperative sentence, but is not either true or false. It doesn’t matter that this might actually be the rule or not. Note that “The rule is that all customers must wear shoes” is a statement. (b) This is a statement, as it is either true or false. It is an atomic statement because it cannot be divided into smaller statements. (c) This is again a statement, but this time it is molecular. In fact, it is a conjunction, as we can write it as “The customers wore shoes and the customers wore socks.” 0.2.3. (a) P ∧ Q. (b) P → ¬Q. (c) Jack passed math or Jill passed math (or both). (d) If Jack and Jill did not both pass math, then Jill did. (e) i. Nothing else. ii. Jack did not pass math either. 0.2.4. (a) It is impossible to tell. The hypothesis of the implication is true. Thus the implication will be true if the conclusion is true (if 13 is my favorite number) and false otherwise. (b) This is true, no matter whether 13 is my favorite number or not. Any implication with a true conclusion is true. (c) This is true, again, no matter whether 13 is my favorite number or not. Any implication with a false hypothesis is true. (d) For a disjunction to be true, we just need one or the other (or both) of the parts to be true. Thus this is a true statement. (e) We cannot tell. The statement would be true if 13 is my favorite number, and false if not (since a conjunction needs both parts to be true to be true). 335 336 B. Selected Solutions (f) This is deﬁnitely false. 13 is prime, so its negation (13 is not prime) is false. At least one part of the conjunction is false, so the whole statement is false. (g) This is true. Either 13 is my favorite number or it is not, but whichever it is, at least one part of the disjunction is true, so the whole statement is true. 0.2.5. The main thing to realize is that we don’t know the colors of these two shapes, but we do know that we are in one of three cases: We could have a blue square and green triangle. We could have a square that was not blue but a green triangle. Or we could have a square that was not blue and a triangle that was not green. The case in which the square is blue but the triangle is not green cannot occur, as that would make the statement false. (a) This must be false. In fact, this is the negation of the original implication. (b) This might be true or might be false. (c) True. This is the contrapositive of the original statement, which is logically equivalent to it. (d) We do not know. This is the converse of the original statement. In particular, if the square is not blue but the triangle is green, then the original statement is true but the converse is false. (e) True. This is logically equivalent to the original statement. 0.2.6. The only way for an implication P → Q to be true but its converse to be false is for Q to be true and P to be false. Thus: (a) False. (b) True. (c) False. (d) True. 0.2.7. The converse is “If I will give you magic beans, then you will give me a cow.” The contrapositive is “If I will not give you magic beans, then you will not give me a cow.” All the other statements are neither the converse nor contrapositive. 0.2.9. (a) Equivalent to the converse. (b) Equivalent to the original theorem. (c) Equivalent to the converse. Selected Solutions 337 (d) Equivalent to the original theorem. (e) Equivalent to the original theorem. (f) Equivalent to the converse. (g) Equivalent to the converse. (h) Equivalent to the original theorem. 0.2.10. (a) If you have lost weight, then you exercised. (b) If you exercise, then you will lose weight. (c) If you are American, then you are patriotic. (d) If you are patriotic, then you are American. (e) If a number is rational, then it is real. (f) If a number is not even, then it is prime. (Or the contrapositive: if a number is not prime, then it is even.) (g) If the Broncos don’t win the Super Bowl, then they didn’t play in the Super Bowl. Alternatively, if the Broncos play in the Super Bowl, then they will win the Super Bowl. 0.2.12. P(5) is the statement “3 · 5 + 1 is even”, which is true. Thus the statement ∃xP(x) is true (for example, 5 is such an x). However, we cannot tell anything about ∀xP(x) since we do not know the truth value of P(x) for all elements of the domain of discourse. In this case, ∀xP(x) happens to be false (since P(4) is false, for example). 0.2.14. (a) The claim that ∀xP(x) means that P(n) is true no matter what n you consider in the domain of discourse. Thus the only way to prove that ∀xP(x) is true is to check or otherwise argue that P(n) is true for all n in the domain. (b) To prove ∀xP(x) is false all you need is one example of an element in the domain for which P(n) is false. This is often called a counterexample. (c) We are simply claiming that there is some element n in the domain of discourse for which P(n) is true. If you can ﬁnd one such element, you have veriﬁed the claim. (d) Here we are claiming that no element we ﬁnd will make P(n) true. The only way to be sure of this is to verify that every element of the domain makes P(n) false. Note that the level of proof needed for this statement is the same as to prove that ∀xP(x) is true. 338 B. Selected Solutions 0.2.15. (a) ∀x∃yP(x, y) is false because when x 4, there is no y which makes P(4, y) true. (b) ∀y∃xP(x, y) is true. No matter what y is (i.e., no matter what column we are in) there is some x for which P(x, y) is true. In fact, we can always take x to be 3. (c) ∃x∀yP(x, y) is true. In particular x 3 is such a number, so that no matter what y is, P(x, y) is true. (d) ∃y∀xP(x, y) is false. In fact, no matter what y (column) we look at, there is always some x (row) which makes P(x, y) false. 0.2.16. (a) ¬∃x(E(x) ∧ O(x)). (b) ∀x(E(x) → O(x + 1)). (c) ∃x(P(x) ∧ E(x)) (where P(x) means “x is prime”). (d) ∀x∀y∃z(x < z < y ∨ y < z < x). (e) ∀x¬∃y(x < y < x + 1). 0.2.17. (a) Any even number plus 2 is an even number. (b) For any x there is a y such that sin(x) y. In other words, every number x is in the domain of sine. (c) For every y there is an x such that sin(x) y. In other words, every number y is in the range of sine (which is false). (d) For any numbers, if the cubes of two numbers are equal, then the numbers are equal. 0.3 Exercises 0.3.1. (a) {1, 3, 4, 6, 9}. (b) {1}. (c) {4, 9}. (d) {3, 6, 10}. 0.3.2. (a) This is the set {3, 4, 5, . . .} since we need each element to be a natural number whose square is at least three more than 2. Since 32 − 3 6 but 22 − 3 1 we see that the ﬁrst such natural number is 3. Selected Solutions 339 (b) We get the same set as we did in the previous part, and the smallest non-negative number for which n 2 − 5 is a natural numbers is 3. Note that if we didn’t specify n ∈ N then any integer less than −3 would also be in the set, so there would not be a least element. (c) This is the set {1, 2, 5, 10, . . .}, namely the set of numbers that are the result of squaring and adding 1 to a natural number. (02 + 1 1, 12 + 1 2, 22 + 1 5 and so on.) Thus the least element of the set is 1. (d) Now we are looking for natural numbers that are equal to taking some natural number, squaring it and adding 1. That is, {1, 2, 5, 10, . . .}, the same set as the previous part. So again, the least element is 1. 0.3.3. (a) 34. Note that 37 − 4 33, but this calculation would not include 4 itself. (b) 103. Again, you could compute this by 100 − (−2) + 1, or simply count: 100 numbers from 1 through 100, plus -2, -1, and 0. (c) 8. There are 8 primes not greater than 20: {2, 3, 5, 7, 11, 13, 17, 19}. 0.3.4. {2, 4}. 0.3.5. {1, 2, 3, 4, 5, 6, 8, 10} 0.3.6. 11. 0.3.7. There will be exactly 4 such sets: {2, 3, 4}, {1, 2, 3, 4}, {2, 3, 4, 5} and {1, 2, 3, 4, 5}. 0.3.8. (a) A ∩ B {3, 4, 5}. (b) A ∪ B {1, 2, 3, 4, 5, 6, 7}. (c) A \ B {1, 2}. (d) A ∩ (B ∪ C) {1}. 0.3.9. (a) A ∩ B will be the set of natural numbers that are both at least 4 and less than 12, and even. That is, A ∩ B {x ∈ N : 4 ≤ x < 12 ∧ x is even} {4, 6, 8, 10}. (b) A \ B is the set of all elements that are in A but not B. So this is {x ∈ N : 4 ≤ x < 12 ∧ x is odd} {5, 7, 9, 11}. Note this is the same set as A ∩ B. 340 B. Selected Solutions 0.3.11. For example, A {2, 3, 5, 7, 8} and B {3, 5}. 0.3.12. For example, A {1, 2, 3} and B {1, 2, 3, 4, 5, {1, 2, 3}} 0.3.13. (a) No. (b) No. (c) 2Z ∩ 3Z is the set of all integers which are multiples of both 2 and 3 (so multiples of 6). Therefore 2Z ∩ 3Z {x ∈ Z : ∃y ∈ Z(x 6y)}. (d) 2Z ∪ 3Z. 0.3.15. (a) A ∪ B: (d) (A ∩ B) ∪ C: A A B B C (b) (A ∪ B): (e) A ∩ B ∩ C: A B A B C (c) A ∩ (B ∪ C): A B (f) (A ∪ B) \ C: A B C C 0.3.17. P(A) {∅, {a}, {b}, {c}, {d}, {a, b}, {a, c}, {a, d}, {b, c}, {b, d}, {c, d}{a, b, c}, {a, b, d}, {a, c, d}, {b, c, d}, {a, b, c, d}}. 0.3.20. For example, A {1, 2, 3, 4} and B {5, 6, 7, 8, 9} gives A ∪ B {1, 2, 3, 4, 5, 6, 7, 8, 9}. 0.3.28. We need to be a little careful here. If B contains 3 elements, then A contains just the number 3 (listed twice). So that would make |A| 1, which would make B {1, 3}, which only has 2 elements. Thus |B| , 3. This means that |A| 2, so B contains at least the elements 1 and 2. Since |B| , 3, we must have |B| 2, which agrees with the deﬁnition of B. Therefore it must be that A {2, 3} and B {1, 2} Selected Solutions 341 0.4 Exercises 0.4.1. (a) f (1) 4, since 4 is the number below 1 in the two-line notation. (b) Such an n is n 2, since f (2) 1. Note that 2 is above a 1 in the notation. (c) n 4 has this property. We say that 4 is a ﬁxed point of f . Not all functions have such a point. (d) Such an element is 2 (in fact, that is the only element in the codomain that is not in the range). In other words, 2 is not the image of any element under f ; nothing is sent to 2. 0.4.2. (a) This is neither injective nor surjective. It is not injective because more than one element from the domain has 3 as its image. It is not surjective because there are elements of the codomain (1, 2, 4, and 5) that are not images of anything from the domain. (b) This is a bĳection. Every element in the codomain is the image of exactly one element of the domain. (c) This is a bĳection. Note that we can write this function in two line 1 2 3 4 5 notation as f . 5 4 3 2 1 1 2 3 4 5 (d) In two line notation, this function is f . From this 1 1 2 2 3 we can quickly see it is neither injective (for example, 1 is the image of both 1 and 2) nor surjective (for example, 4 is not the image of anything). 0.4.5. There are 8 functions, including 6 surjective and zero injective funtions. 0.4.7. (a) f is not injective, since f (2) f (5); two diﬀerent inputs have the same output. (b) f is surjective, since every element of the codomain is an element of the range. 1 2 3 4 5 (c) f . 3 2 4 1 2 0.4.9. f (10) 1024. To ﬁnd f (10), we need to know f (9), for which we need f (8), and so on. So build up from f (0) 1. Then f (1) 2, f (2) 4, 342 B. Selected Solutions f (3) 8, .... In fact, it looks like a closed formula for f is f (n) 2n . Later we will see how to prove this is correct. 0.4.10. For each case, you must use the recurrence to ﬁnd f (1), f (2) ... f (5). But notice each time you just add three to the previous. We do this 5 times. (a) f (5) 15. (b) f (5) 16. (c) f (5) 17. (d) f (5) 115. 0.4.12. (a) f is injective, but not surjective (since 0, for example, is never an output). (b) f is injective and surjective. Unlike in the previous question, every integers is an output (of the integer 4 less than it). (c) f is injective, but not surjective (10 is not 8 less than a multiple of 5, for example). (d) f is not injective, but is surjective. Every integer is an output (of twice itself, for example) but some integers are outputs of more than one input: f (5) 3 f (6). 0.4.13. (a) f is not injective. To prove this, we must simply ﬁnd two diﬀerent elements of the domain which map to the same element of the codomain. Since f ({1}) 1 and f ({2}) 1, we see that f is not injective. (b) f is not surjective. The largest subset of A is A itself, and |A| 10. So no natural number greater than 10 will ever be an output. (c) f −1 (1) {{1}, {2}, {3}, . . . {10}} (the set of all the singleton subsets of A). (d) f −1 (0) {∅}. Note, it would be wrong to write f −1 (0) ∅ - that would claim that there is no input which has 0 as an output. (e) f −1 (12) ∅, since there are no subsets of A with cardinality 12. 0.4.16. (a) | f −1 (3)| ≤ 1. In other words, either f −1 (3) is the empty set or is a set containing exactly one element. Injective functions cannot have two elements from the domain both map to 3. Selected Solutions 343 (b) | f −1 (3)| ≥ 1. In other words, f −1 (3) is a set containing at least one elements, possibly more. Surjective functions must have something map to 3. (c) | f −1 (3)| 1. There is exactly one element from X which gets mapped to 3, so f −1 (3) is the set containing that one element. 0.4.17. X can really be any set, as long as f (x) 0 or f (x) 1 for every x ∈ X. For example, X N and f (n) 0 works. 0.4.21. (a) f is injective. Proof. Let x and y be elements of the domain Z. Assume f (x) f (y). If x and y are both even, then f (x) x + 1 and f (y) y + 1. Since f (x) f (y), we have x+1 y+1 which implies that x y. Similarly, if x and y are both odd, then x − 3 y − 3 so again x y. The only other possibility is that x is even an y is odd (or visa-versa). But then x + 1 would be odd and y − 3 would be even, so it cannot be that f (x) f (y). Therefore if f (x) f (y) we then have x y, which proves that f is injective. qed (b) f is surjective. Proof. Let y be an element of the codomain Z. We will show there is an element n of the domain (Z) such that f (n) y. There are two cases: First, if y is even, then let n y + 3. Since y is even, n is odd, so f (n) n − 3 y + 3 − 3 y as desired. Second, if y is odd, then let n y − 1. Since y is odd, n is even, so f (n) n + 1 y − 1 + 1 y as needed. Therefore f is surjective. qed 0.4.22. Yes, this is a function, if you choose the domain and codomain correctly. The domain will be the set of students, and the codomain will be the set of possible grades. The function is almost certainly not injective, because it is likely that two students will get the same grade. The function might be surjective – it will be if there is at least one student who gets each grade. 0.4.24. This is not a function. 0.4.25. The recurrence relation is f (n + 1) f (n) + n. 0.4.26. In general, |A| ≥ f (A) , since you cannot get more outputs than you have inputs (each input goes to exactly one output), but you could have fewer outputs if the function is not injective. If the function is injective, then |A| f (A) , although you can have equality even if f is not injective (it must be injective restricted to A). 344 B. Selected Solutions 0.4.27. In general, there is no relationship between |B| and f −1 (B) . This is because B might contain elements that are not in the range of f , so we might even have f −1 (B) ∅. On the other hand, there might be lots of elements from the domain that all get sent to a few elements in B, making f −1 (B) larger than B. More speciﬁcally, if f is injective, then |B| ≥ f −1 (B) (since every element in B must come from at most one element from the domain). If f is surjective, then |B| ≤ f −1 (B) (since every element in B must come from at least one element of the domain). Thus if f is bĳective then |B| f −1 (B) . 1.1 Exercises 1.1.1. There are 255 outﬁts. Use the multiplicative principle. 1.1.2. (a) 8 ties. Use the additive principle. (b) 15 ties. Use the multiplicative principle (c) 5 · (4 + 3) + 7 42 outﬁts. 1.1.3. (a) For example, 16 is the number of choices you have if you want to watch one movie, either a comedy or horror ﬂick. (b) For example, 63 is the number of choices you have if you will watch two movies, ﬁrst a comedy and then a horror. 1.1.5. (a) To maximize the number of elements in common between A and B, make A ⊂ B. This would give |A ∩ B| 10. (b) A and B might have no elements in common, giving |A ∩ B| 0. (c) 15 ≤ |A ∪ B| ≤ 25. In fact, when |A ∩ B| 0 then |A ∪ B| 25 and when |A ∩ B| 10 then |A ∪ B| 15. 1.1.6. |A ∪ B| + |A ∩ B| 13. Use PIE: we know |A ∪ B| 8 + 5 − |A ∩ B| . 1.1.7. 39 students. Use Venn diagram or PIE: 28+19+24−16−14−10+8 39. 1.1.8. 6 students don’t like potatoes. 1.1.9. 215 values of n. 1.1.12. (a) 85 32768 words. (b) 8 · 7 · 6 · 5 · 4 6720 words. Selected Solutions 345 (c) 8 · 8 64 words. (d) 64 + 64 − 0 128 words. (e) (8 · 7 · 6 · 5 · 4) − 3 · (5 · 4) 6660 words. 1.2 Exercises 1.2.1. (a) 26 64 subsets. We need to select yes/no for each of the six elements. (b) 23 8 subsets. We need to select yes/no for each of the remaining three elements. (c) 26 − 23 56 subsets. There are 8 subsets which do not contain any odd numbers (select yes/no for each even number). (d) 3 · 23 24 subsets. First pick the even number. Then say yes or no to each of the odd numbers. 1.2.2. (a) (b) (c) (d) 6 4 3 15 subsets. 3 subsets. We need to select 1 of the 3 remaining elements to be in the subset. 1 6 4 15 subsets. All subsets of cardinality 4 must contain at least one odd number. 3 1 3 subsets. Select 1 of the 3 even numbers. The remaining three odd numbers of S must all be in the set. 1.2.3. (a) There are 512 subsets. (b) 9 5 126. (c) 24 16. (Note, if you wish to exclude the empty set - it does not contain odd numbers, but no evens either - then you could subtract 1). (d) 256. 1.2.4. (a) 26 64. (b) 6 3 20. (c) 176. (d) 51. 346 B. Selected Solutions 1.2.5. (a) We will need 6 · 20 120 coins (60 of each). (b) We need 6 · 64 384 coins (192 of each). 10 6 1.2.6. 14 9 1.2.9. 1.2.10. (a) 14 7 (b) 2 14 7 1.2.11. (a) 1.2.12. (a) (b) (c) 5 10 9 + 10 10 386 strings. 840 paths. 6 8 2 5 − − 3 15 1 8 (d) + paths. 18 9 . 18 9 (c) 10 8 3432 paths. 12 6 7 2 . (b) + 15 9 6 2 . + 6 8 (c) 10 7 + 3 14 1 8 + 12 6 7 2 − 3 9 6 1 6 2 11 3 165 choices, since you have to select a 3-element subset of the set of 11 toppings. 10 3 120 choices, since you must select 3 of the 10 non-pineapple toppings. 10 2 45 choices, since you must select 2 of the remaining 10 nonpineapple toppings to have in addition to the pineapple. (d) 165 120 + 45 choices, which makes sense because every 3-topping pizza either has pineapple or does not have pineapple as a topping. Selected Solutions 347 1.3 Exercises 1.3.1. (a) 10 3 120 pizzas. We must choose (in no particular order) 3 out of the 10 toppings. (b) 210 1024 pizzas. Say yes or no to each topping. (c) P(10, 5) 30240 ways. Assign each of the 5 spots in the left column to a unique pizza topping. 1.3.2. Despite its name, we are not looking for a combination here. The order in which the three numbers appears matters. There are P(40, 3) 40 · 39 · 38 diﬀerent possibilities for the “combination”. This is assuming you cannot repeat any of the numbers (if you could, the answer would be 403 ). 1.3.3. (a) This is just the multiplicative principle. There are 7 digits which we can select for each of the 5 positions, so we have 75 16807 such numbers. (b) Now we have 7 choices for the ﬁrst number, 6 for the second, etc. So there are 7 · 6 · 5 · 4 · 3 P(7, 5) 2520 such numbers. (c) To build such a number we simply must select 5 diﬀerent digits. After doing so, there will only be one way to arrange them into a 5-digit number. Thus there are 75 21 such numbers. (d) The permutation is in part (b), while the combination is in part (c). At ﬁrst this seems backwards, since usually we use combinations for when order does not matter. Here it looks like in part (c) that order does matter. The better way to distinguish between combinations and permutations is to ask whether we are counting diﬀerent arrangements as diﬀerent outcomes. In part (c), there is only one arrangement of any set of 5 digits, while in part (b) each set of 5 digits gives 5! diﬀerent outcomes. 1.3.5. You can make 72 72 441 quadrilaterals. There are 5 squares. There are 72 rectangles. There are 72 + ( 72 − 1) + ( 72 − 3) + ( 72 − 6) + ( 72 − 10) + ( 72 − 15) 91 parallelograms. All of the quadrilaterals are trapezoids. To count the non-parallelogram trapezoids, compute 441 − 91 350. 1.3.6. 120. 348 B. Selected Solutions 1.3.7. Since there are 15 diﬀerent letters, we have 15 choices for the ﬁrst letter, 14 for the next, and so on. Thus there are 15! anagrams. 1.3.8. There are 7 2 21 anagrams starting with “a”. 1.3.9. First, decide where to put the “a”s. There are 7 positions, and we must choose 3 of them to ﬁll with an “a”. This can be done in 73 ways. The remaining 4 spots all get a diﬀerent letter, so there are 4! ways to ﬁnish oﬀ the anagram. This gives a total of 73 · 4! anagrams. Strangely enough, this is 840, which is also equal to P(7, 4). To get the answer that way, start by picking one of the 7 positions to be ﬁlled by the “n”, one of the remaining 6 positions to be ﬁlled by the “g”, one of the remaining 5 positions to be ﬁlled by the “r”, one of the remaining 4 positions to be ﬁlled by the “m” and then put “a”s in the remaining 3 positions. 1.3.10. (a) 20 16 12 8 4 4 4 4 4 4 (b) 5! ways. 15 12 9 6 3 3 3 3 3 3 ways. 1.3.11. 9!. 1.3.12. (a) 1710 functions. There are 17 choices for the image of each element in the domain. (b) P(17, 10) injective functions. There are 17 choices for image of the ﬁrst element of the domain, then only 16 choices for the second, and so on. 1.3.13. (a) 64 1296 functions. (b) P(6, 4) 6 · 5 · 4 · 3 360 functions. (c) 6 4 15 functions. Selected Solutions 349 1.4 Exercises 1.4.1. Proof. Question: How many 2-letter words start with a, b, or c and end with either y or z? Answer 1: There are two words that start with a, two that start with b, two that start with c, for a total of 2 + 2 + 2. Answer 2: There are three choices for the ﬁrst letter and two choices for the second letter, for a total of 3 · 2. Since the two answers are both answers to the same question, they are equal. Thus 2 + 2 + 2 3 · 2. qed 1.4.5. (a) She has 15 for each way, 6 ways to select the 6 bridesmaids, and then 15 has 6 choices for the maid of honor. Thus she has 6 6 choices. (b) She has 15 choices for who will be her maid of honor. Then she needs to select 5 of the remaining 14 friends to be bridesmaids, which she 14 can do in 14 ways. Thus she has 15 choices. 5 5 (c) We have answered the question (how many wedding parties can the bride choose from) in two ways. The ﬁrst way gives the left-hand side of the identity and the second way gives the right-hand side of the identity. Therefore the identity holds. 1.4.7. Proof. Question: You have a large container ﬁlled with ping-pong balls, all with a diﬀerent number on them. You must select k of the balls, putting two of them in a jar and the others in a box. How many ways can you do this? Answer 1: First select 2 of the n balls to put in the jar. Then select k − 2 of the remaining n − 2 balls to put in the box. The ﬁrst task can be n n−2 completed in 2 diﬀerent ways, the second task in k−2 ways. Thus there are n2 n−2 k−2 ways to select the balls. Answer 2: First select k balls from the n in the container. Then pick 2 of the k balls you picked to put in the jar, placing the remaining k − 2 in the box. The ﬁrst task can be completed in nk ways, the second task in 2k ways. Thus there are nk 2k ways to select the balls. Since both answers count the same thing, they must be equal and the identity is established. qed 1.5 Exercises 1.5.1. (a) 10 5 sets. We must select 5 of the 10 digits to put in the set. 350 B. Selected Solutions (b) Use stars and bars: each star represents one of the 5 elements of the set, each bar represents a switch between digits. So there are 5 stars and 9 bars, giving us 14 9 sets. 1.5.2. (a) There are 75 numbers. We simply choose ﬁve of the seven digits and once chosen put them in increasing order. (b) This requires stars and bars. Use a star to represent each of the 5 digits in the number, and use their position relative to the bars to say what numeral ﬁlls that spot. So we will have 5 stars and 6 bars, giving 11 numbers. 6 1.5.3. (a) You take 3 strawberry, 1 lime, 0 licorice, 2 blueberry and 0 bubblegum. (b) This is backwards. We don’t want the stars to represent the kids because the kids are not identical, but the stars are. Instead we should use 5 stars (for the lollipops) and use 5 bars to switch between the 6 kids. For example, ∗ ∗ || ∗ ∗ ∗ ||| would represent the outcome with the ﬁrst kid getting 2 lollipops, the third kid getting 3, and the rest of the kids getting none. (c) This is the word AAAEOO. (d) This doesn’t represent a solution. Each star should represent one of the 6 units that add up to 6, and the bars should switch between the diﬀerent variables. We have one too many bars. An example of a correct diagram would be ∗| ∗ ∗|| ∗ ∗ ∗ , representing that x 1 1, x 2 2, x 3 0, and x 4 3. 1.5.4. (a) (b) 18 4 ways. Each outcome can be represented by a sequence of 14 stars and 4 bars. 13 4 ways. First put one ball in each bin. This leaves 9 stars and 4 bars. Selected Solutions 1.5.5. (a) 7 2 solutions. After each variable gets 1 star for free, we are left with 5 stars and 2 bars. 10 2 (b) (c) 351 19 solutions. We have 8 stars and 2 bars. solutions. This problem is equivalent to ﬁnding the number of solutions to x ′ + y ′ + z ′ 17 where x ′, y ′ and z ′ are non-negative. (In fact, we really just do a substitution. Let x x ′ − 3, y y ′ − 3 and z z ′ − 3). 2 10 5 1.5.6. outcomes. 1.5.7. There are 10 3 120 diﬀerent combinations of coins possible. Thus you have a 1 in 120 chance of guessing correctly. 18 1.5.8. 15 solutions. Distribute 10 units to the variables before ﬁnding all solutions to x1′ + x2′ + x 3′ + x 4′ 15 in non-negative integers. 1.5.10. (a) 14 9 . (b) 1.5.11. (a) 10 5 . Note that a strictly increasing function is automatically injective. 20 4 sodas (order does not matter and repeats are not allowed). (b) P(20, 4) 20 · 19 · 18 · 17 sodas (order matters and repeats are not allowed). (c) 23 4 sodas (order does not matter and repeats are allowed; 4 stars and 19 bars). (d) 204 sodas (order matters and repeats are allowed; 20 choices 4 times). 1.6 Exercises 1.6.1. (a) (b) (c) 1.6.3. 9 3 meals. First spend $7 to buy one of each item, then use 3 stars to select items between 6 bars. 16 6 meals. Here you have 10 stars and 6 bars (separating the 7 items). 16 6 7 10 7 7 − 17 13 meals. Use PIE to subtract all the 6 − 2 6 + 3 6 meals in which you get 3 or more of a particular item. 18 4 − 5 1 1.6.4. There are 11 4 10 7 − − 5 4 2 4 4 6 1 3 . solutions. 352 B. Selected Solutions 1.6.5. Without any restriction, there would be 19 7 ways to distribute the stars. Now we must use PIE to eliminate all distributions in which one or more student gets more than one star: 19 − 7 13 17 13 15 13 13 − + 1 5 2 3 3 1 1716. Interestingly enough, this number is also the value of 13 7 , which makes sense: if each student can have at most one star, we must just pick the 7 out of 13 students to receive them. 1.6.7. The 9 derangements are: 2143, 2341, 2413, 3142, 3412, 3421, 4123, 4312, 4321. 1.6.8. First pick one of the ﬁve elements to be ﬁxed. For each such choice, derange the advanced PIE formula. remaining four, using the standard 5 1 We get 1.6.9. 10 6 4! − 4! − 4 1 3! − 4 1 4 2 2! + 4 3 1! − 4 4 0! 4 2 2! + 4 3 1! − 4 4 0! 3! − permutations. ways. We choose 6 of the 10 ladies to get their own hat, and the multiply by the number of ways the remaining hats can be deranged. 1.6.11. There are 5 · 63 functions for which f (1) , a and another 5 · 63 functions for which f (2) , b. There are 52 · 62 functions for which both f (1) , a and f (2) , b. So the total number of functions for which f (1) , a or f (2) , b or both is 5 · 63 + 5 · 63 − 52 · 62 1260. 5 5 0! 1.6.12. 510 − 51 410 − 52 310 + 53 210 − 54 110 functions. The 510 is all the functions from A to B. We subtract those that aren’t surjective. Pick one of the ﬁve elements in B to not have in the range (in 51 ways) and count all those functions (410 ). But this overcounts the functions where two elements from B are excluded from the range, so subtract those. And so on, using PIE. 1.6.13. 5! − 5 1 4! − 5 2 3! + 5 3 2! − 5 4 1! + functions. 1.7 Chapter Review 1.7.1. (a) (b) 8 5 ways, after giving one present to each kid, you are left with 5 presents (stars) which need to be divide among the 4 kids (giving 3 bars). 12 9 ways. You have 9 stars and 3 bars. Selected Solutions 353 (c) 49 . You have 4 choices for whom to give each present. This is like making a function from the set of presents to the set of kids. (d) 49 − 41 39 − 42 29 + 43 19 ways. Now the function from the set of presents to the set of kids must be surjective. 1.7.2. (a) Neither. (b) 10 4 14 4 paths. bow ties. (c) P(10, 4), since order is important. (d) Neither. Assuming you will wear each of the 4 ties on just 4 of the 7 days, without repeats: 10 4 P(7, 4). (e) P(10, 4). (f) 10 4 . (g) Neither. Since you could repeat letters: 104 . If no repeats are allowed, it would be P(10, 4). (h) Neither. Actually, “k” is the 11th letter of the alphabet, so the answer is 0. If “k” was among the ﬁrst 10 letters, there would only be 1 way - write it down. (i) Neither. Either 93 (if every kid gets an apple) or kids are allowed). 13 3 (if appleless (j) Neither. Note that this could not be 10 4 since the 10 things and 4 10 things are from diﬀerent groups. 4 , assuming each kid eats one type of cerial. (k) (l) 10 4 - don’t be fooled by the “arrange” in there - you are picking 4 out of 10 spots to put the 1’s. 10 4 (assuming order is irrelevant). (m) Neither. 1610 (each kid chooses yes or no to 4 varieties). (n) Neither. 0. (o) Neither. 410 − [ 4 10 1 3 − 4 10 2 2 + 4 10 3 1 ]. (p) Neither. 10 · 4. (q) Neither. 410 . (r) 10 4 (which is the same as 10 6 ). (s) Neither. If all the kids were identical, and you wanted no empty 10 teams, it would be 4 . Instead, this will be the same as the number of surjective functions from a set of size 11 to a set of size 5. 354 (t) (u) B. Selected Solutions 10 4 . 10 4 . (v) Neither. 4!. (w) Neither. It’s 10 4 if you won’t repeat any choices. If repetition is allowed, then this becomes x 1 + x 2 + · · · + x 10 4, which has 13 9 solutions in non-negative integers. (x) Neither. Since repetition of cookie type is allowed, the answer is 104 . Without repetition, you would have P(10, 4). (y) 10 4 since that is equal to 9 4 + 9 3 . (z) Neither. It will be a complicated (possibly PIE) counting problem. 1.7.3. (a) 28 256 choices. You have two choices for each tie: wear it or don’t. (b) You have 7 choices for regular ties (the 8 choices less the “no regular tie” option) and 31 choices for bow ties (32 total minus the “no bow tie” option). Thus total you have 7 · 31 217 choices. (c) 3 5 2 3 30 choices. (d) Select one of the 3 bow ties to go on top. There are then 4 choices for the next tie, 3 for the tie after that, and so on. Thus 3 · 4! 72 choices. 1.7.4. You own 8 purple bow ties, 3 red bow ties, 3 blue bow ties and 5 green bow ties. How many ways can you select one of each color bow tie to take with you on a trip? 8 · 3 · 3 · 5 ways. How many choices do you have for a single bow tie to wear tomorrow? 8 + 3 + 3 + 5 choices. 1.7.5. (a) 45 numbers. (b) 44 · 2 numbers (choose any digits for the ﬁrst four digits - then pick either an even or an odd last digit to make the sum even). (c) We need 3 or more even digits. 3 even digits: 53 23 22 . 4 even digits: 5 4 5 5 5 3 2 5 4 5 5 4 2 2. 5 even digits: 5 2 . So all together: 3 2 2 + 4 2 2 + 5 2 numbers. 1.7.6. 51 passengers. We are asking for the size of the union of three non-disjoint sets. Using PIE, we have 25 + 30 + 20 − 10 − 12 − 7 + 5 51. 1.7.7. (a) 28 strings. (b) 8 5 strings. Selected Solutions (c) 8 5 355 strings. (d) There is a bĳection between subsets and bit strings: a 1 means that element in is the subset, a 0 means that element is not in the subset. To get a subset of an 8 element set we have a 8-bit string. To make sure the subset contains exactly 5 elements, there must be 5 1’s, so the weight must be 5. 13 10 1.7.8. 17 8 . + 1.7.9. With repeated letters allowed, we select which 5 of the 8 letters will be vowels, then pick one of the 5 vowels for each spot, and ﬁnally pick one of the other 21 letters for each of the remaining 3 spots. Thus, 8 5 3 5 21 words. 5 Without repeats, we still pick the positions of the vowels, but now each time we place one there, there is one fewer choice for the next one. 8 Similarly, we cannot repeat the consonants. We get 5 5!P(21, 3) words. 1.7.10. (a) 5 11 2 6 (b) 8 (c) 16 paths. 5 − 11 12 4 7 1 2 6 + 1.7.11. 18 8 paths. 12 4 5 3 18 8 − 5 7 4 2 3 3 paths. − 1 routes. For each of the 18 8 routes to work there is exactly one fewer route back. 1.7.12. 27 + 27 − 24 strings (using PIE). 7 3 1.7.13. + 7 4 − 4 1 strings. 1.7.14. There are 4 spots to start the word, and then there are 3! ways to arrange the other letters in the remaining three spots. Thus the number of words avoiding the sub-word “bad” in consecutive letters is 6! − 4 · 3!. If we now need to avoid words that put “b” before “a” before “d”, we must choose which spots those letters go (in that order) and then arrange the remaining three letters. Thus, 6! − 63 3! words. 1.7.15. 2n is the number of lattice paths which have length n, since for each step you can go up or right. Such a path would end along the line x + y n. So you will end at (0, n), or (1, n − 1) or (2, n − 2) or . . . or n n (n, 0). Counting the paths to each of these points separately, give , 0 1 , n n 2 , . . . , n (each time choosing which of the n steps to be to the right). These two methods count the same quantity, so are equal. 1.7.16. (a) 19 15 ways. 356 B. Selected Solutions (b) 24 20 ways. (c) 15 − 19 5 12 8 1 − 5 5 2 1 ways. 1.7.17. (a) 54 + 54 − 53 functions. (b) 4 · 54 + 5 · 4 · 53 − 4 · 4 · 53 functions. (c) 5! − [4! + 4! − 3!] functions. Note we use factorials instead of powers because we are looking for injective functions. (d) Note that being surjective here is the same as being injective, so we can start with all 5! injective functionsand subtract those which have 5 5 5 5 one or more “ﬁxed point”. We get 5!− 1 4! − 2 3! + 3 2! − 4 1! + 55 0! functions. 1.7.18. 46 − 1.7.19. (a) 4 1 36 − 4 6 2 2 + 4 6 3 1 . 10 4 combinations. You need to choose 4 of the 10 cookie types. Order doesn’t matter. (b) P(10, 4) 10 · 9 · 8 · 7 ways. You are choosing and arranging 4 out of 10 cookies. Order matters now. (c) 21 12 choices. You must switch between cookie type 9 times as you make your 12 cookies. The cookies are the stars, the switches between cookie types are the bars. (d) 1012 choices. You have 10 choices for the “1” cookie, 10 choices for the “2” cookie, and so on. 10 12 10 12 12 choices. We must use PIE to (e) 1012 − 10 1 9 − 2 8 + · · · − 10 0 remove all the ways in which one or more cookie type is not selected. 1.7.20. (a) You are giving your professor 4 types of cookies coming from 10 diﬀerent types of cookies. This does not lend itself well to a function interpretation. We could say that the domain contains the 4 types you will give your professor and the codomain contains the 10 you can choose from, but then counting injections would be too much (it doesn’t matter if you pick type 3 ﬁrst and type 2 second, or the other way around, just that you pick those two types). (b) We want to consider injective functions from the set {most, second most, second least, least} to the set of 10 cookie types. We want injections because we cannot pick the same type of cookie to give most and least of (for example). Selected Solutions 357 (c) This is not a good problem to interpret as a function. The problem is that the domain would have to be the 12 cookies you bake, but these elements are indistinguishable (there is not a ﬁrst cookie, second cookie, etc.). (d) The domain should be the 12 shapes, the codomain the 10 types of cookies. Since we can use the same type for diﬀerent shapes, we are interested in counting all functions here. (e) Here we insist that each type of cookie be given at least once, so now we are asking for the number of surjections of those functions counted in the previous part. 2.1 Exercises 2.1.1. (a) Note that if we subtract 1 from each term, we get the familiar square numbers. Thus a n n 2 + 1. (b) These look like the triangular numbers, only shifted by 1. We get: n(n+1) a n 2 − 1. (c) If you subtract 2 from each term, you get triangular numbers, only starting with 6 instead of 1. So we must shift vertically and horizon(n+2)(n+3) + 2. tally. a n 2 (d) These seem to grow very quickly. Further, if we add 1 to each term, we ﬁnd the factorials, although starting with 2 instead of 1. This gives, a n (n + 1)! − 1 (where n! 1 · 2 · 3 · · · n). 2.1.3. (a) a 0 0, a 1 1, a 2 3, a 3 6 a 4 10. The sequence was described by a closed formula. These are the triangular numbers. A recursive deﬁnition is: a n a n−1 + n with a 0 0. (b) This is a recursive deﬁnition. We continue a 2 2, a 3 3, a 4 4, a 5 5, and so on. A closed formula is a n n. (c) We have a 0 1, a 1 1, a 2 2, a 3 6, a 4 24, a 5 120, and so on. The closed formula is a n n!. 2.1.4. (a) The recursive deﬁnition is a n a n−1 + 2 with a 1 1. A closed formula is a n 2n − 1. (b) The sequence of partial sums is 1, 4, 9, 16, 25, 36, . . .. A recursive deﬁnition is (as always) b n b n−1 + a n which in this case is b n b n−1 + 2n − 1. It appears that the closed formula is b n n 2 358 B. Selected Solutions 2.1.5. (a) 0, 1, 2, 4, 7, 12, 20, . . .. (b) F0 + F1 + · · · + Fn Fn+2 − 1. 2.1.6. The sequences all have the same recurrence relation: a n a n−1 + a n−2 (the same as the Fibonacci numbers). The only diﬀerence is the initial conditions. 2.1.7. 3, 10, 24, 52, 108, . . .. The recursive deﬁnition for 10, 24, 52, . . . is a n 2a n−1 + 4 with a 1 10. 2.1.8. −1, −1, 1, 5, 11, 19, . . . Thus the sequence 0, 2, 6, 12, 20, . . . has closed formula a n (n + 1)2 − 3(n + 1) + 2. 2.1.9. This closed formula would have a n−1 3 · 2n−1 + 7 · 5n−1 and a n−2 3 · 2n−2 + 7 · 5n−2 . Then we would have 7a n−1 − 10a n−2 7(3 · 2n−1 + 7 · 5n−1 ) − 10(3 · 2n−2 + 7 · 5n−2 ) 21 · 2n−1 + 49 · 5n−1 − 30 · 2n−2 − 70 · 5n−2 ) 21 · 2n−1 + 49 · 5n−1 − 15 · 2n−1 − 14 · 5n−1 ) 6 · 2n−1 + 35 · 5n−1 3 · 2n + 7 · 5n a n . So the closed formula agrees with the recurrence relation. The closed formula has initial terms a 0 10 and a 1 41. 2.1.13. (a) n Õ (d) 2k. k1 (b) 2k. k1 107 Õ (1 + 4(k − 1)). (e) k1 (c) n Ö 100 Ö k1 50 Õ k1 k . k+1 1 . k 2.1.14. (a) 100 Õ (3 + 4k) 7 + 11 + 15 + · · · + 403. k1 (b) n Õ 2k 1 + 2 + 4 + 8 + · · · + 2n . k0 (c) 50 Õ k2 (k 2 1 1 1 1 1 1+ + + +···+ . 3 8 15 2499 − 1) Selected Solutions (d) 100 Ö k2 4 9 16 10000 · · ··· . 2 9999 (k − 1) 3 8 15 k2 (e) n Ö 359 (2 + 3k) (2)(5)(8)(11)(14) · · · (2 + 3n). k0 2.2 Exercises 2.2.1. (a) a n a n−1 + 4 with a 1 5. (b) a n 5 + 4(n − 1). (c) Yes, since 2013 5 + 4(503 − 1) (so a 503 2013). (d) 133 (e) 538·133 2 35777. (f) b n 1 + (4n+6)n . 2 2.2.2. (a) 32, which is 26 + 6. (b) The sequence is arithmetic, with a0 8 and constant diﬀerence 6, so a n 8 + 6n. (c) 30500. We want 8 + 14 + · · · + 602. Reverse and add to get 100 sums of 610, a total of 61000, which is twice the sum we are looking for. 2.2.3. (a) 36. (b) 253·36 2 4554. 2.2.4. (a) n + 2 terms, since to get 1 using the formula 6n + 7 we must use n −1. Thus we have n terms, plus two, when n 0 and n −1. (b) 6n + 1, which is 6 less than 6n + 7 (or plug in n − 1 for n). (c) (6n+8)(n+2) . Reverse and add. 2 there are n + 2 terms. Each sum gives the constant 6n + 8 and 2.2.5. 68117. 2.2.6. 5·3 2 −5 . Let the sum be S, and compute S − 3S −2S, which causes terms except 5 and −5 · 321 to cancel. Then solve for S. 21 31 2.2.7. 1+ 231 3 5/3 . This time compute S + 23 S. 360 B. Selected Solutions 2.2.8. For arithmetic: x 55/3, y 29/3. For geometric: x 9 and y 3. 2.2.9. For arithmetic: x 14, y 23. For geometric: x 5 ∗ (32/5)( 1/3) and y 5 ∗ (32/5)( 2/3). 2.2.11. We have 2 2, 7 2 + 5, 15 2 + 5 + 8, 26 2 + 5 + 8 + 11, and so on. The terms in the sums are given by the arithmetic sequence Í b n 2 + 3n. In other words, a n nk0 (2 + 3k). To ﬁnd the closed formula, (4+3n)(n+1) we reverse and add. We get a n (we have n + 1 there because 2 there are n + 1 terms in the sum for a n ). 2.3 Exercises 2.3.1. a n n 2 + n. Here we know that we are looking for a quadratic because the second diﬀerences are constant. So a n an 2 + bn + c. Since a 0 0, we know c 0. So just solve the system 2a+b 6 4a + 2b 2.3.2. a n 16 (n 3 + 5n + 6). 2.3.3. a n 16 (n 3 + 6n 2 + 11n + 12). 2.3.4. a n 16 (n 3 + 6n 2 + 11n + 18). 2.3.6. a n n 2 − n + 1. 2.3.7. a n n 3 + n 2 − n + 1 2.3.8. a n−1 (n − 1)2 + 3(n − 1) + 4 n 2 + n + 2. Thus a n − a n−1 2n + 2. Note that this is linear (arithmetic). We can check that we are correct. The sequence a n is 4, 8, 14, 22, 32, . . . and the sequence of diﬀerences is thus 4, 6, 8, 10, . . . which agrees with 2n + 2 (if we start at n 1). 2.3.9. a n−1 a(n − 1)2 + b(n − 1) + c an 2 − 2an + a + bn − b + c. Therefore a n − a n−1 2an − a + b, which is arithmetic. Notice that this is not quite the derivative of a n , which would be 2an + b, but it is close. 2.3.10. No. The sequence of diﬀerences is the same as the original sequence so no diﬀerences will be constant. 2.3.11. No. The sequence is geometric, and in fact has closed formula 2·3n . This is an exponential function, which is not equal to any polynomial of any degree. If the nth sequence of diﬀerences was constant, then the closed formula for the original sequence would be a degree n polynomial. 2.4 Exercises Selected Solutions 361 2.4.1. 171 and 341. a n a n−1 + 2a n−2 with a 0 3 and a 1 5. Closed formula: a n 38 2n + 13 (−1)n . To ﬁnd this solve the characteristic equation, x 2 − x − 2 0, to get characteristic roots x 2 and x −1. Then solve the system 3a+b 5 2a − b 2.4.3. a n 3 + 2n+1 . We should use telescoping or iteration here. For example, telescoping gives a 1 − a 0 21 a 2 − a 1 22 a 3 − a 2 23 .. .. .. a n − a n−1 2n which sums to a n − a 0 2n+1 − 2 (using the multiply-shift-subtract technique from Section 3.2 for the right-hand side). Substituting a 0 5 and solving for a n completes the solution. 2.4.4. We claim a n 4n works. Plug it in: 4n 3(4n−1 ) + 4(4n−2 ). This works - just simplify the right-hand side. 2.4.5. By the Characteristic Root Technique. a n 4n + (−1)n . 2.4.6. a n 13 n 54 + 12 n 5 (−1) . 2.4.7. a n 19 n 7 (−2) + 97 5n . 2.4.10. (a) a n 4a n−1 + 5a n−2 . (b) 4, 21, 104, 521, 2604, 13021 (c) a n 65 5n + 16 (−1)n . 2.4.12. We have characteristic polynomial x 2 − 2x + 1, which has x 1 as the only repeated root. Thus using the characteristic root technique for repeated roots, the general solution is a n a + bn where a and b depend on the initial conditions. (a) a n 1 + n. (b) For example, we could have a 0 21 and a 1 22. (c) For every x. Take a 0 x − 9 and a 1 x − 8. 362 B. Selected Solutions 2.5 Exercises 2.5.1. (a) If we have a number of beans ending in a 5 and we double it, we will get a number of beans ending in a 0 (since 5 · 2 10 ). Then if we subtract 5, we will once again get a number of beans ending in a 5. Thus if on any day we have a number ending in a 5, the next day will also have a number ending in a 5. (b) If you don’t start with a number of beans ending in a 5 (on day 1), the above reasoning is still correct but not helpful. For example, if you start with a number ending in a 3, the next day you will have a number ending in a 1. (c) Part (b) is the base case and part (a) is the inductive case. If on day 1 we have a number ending in a 5 (by part (b)), then on day 2 we will also have a number ending in a 5 (by part (a)). Then by part (a) again, we will have a number ending in a 5 on day 3. By part (a) again, this means we will have a number ending in a 5 on day 4 The proof by induction would say that on every day we will have a number ending in a 5, and this works because we can start with the base case, then use the inductive case over and over until we get up to our desired n. 2.5.2. Proof. We must prove that 1 + 2 + 22 + 23 + · · · + 2n 2n+1 − 1 for all n ∈ N. Thus let P(n) be the statement 1 + 2 + 22 + · · · + 2n 2n+1 − 1. We will prove that P(n) is true for all n ∈ N. First we establish the base case, P(0), which claims that 1 20+1 − 1. Since 21 − 1 2 − 1 1, we see that P(0) is true. Now for the inductive case. Assume that P(k) is true for an arbitrary k ∈ N. That is, 1 + 2 + 22 + · · · + 2k 2k+1 − 1. We must show that P(k + 1) is true (i.e., that 1 + 2 + 22 + · · · + 2k+1 2k+2 − 1). To do this, we start with the left-hand side of P(k + 1) and work to the right-hand side: 1 + 2 + 22 + · · · + 2k + 2k+1 2k+1 − 1 + 2k+1 by inductive hypothesis 2 · 2k+1 − 1 2k+2 − 1 Thus P(k + 1) is true so by the principle of mathematical induction, P(n) is true for all n ∈ N. qed Selected Solutions 363 2.5.3. Proof. Let P(n) be the statement “7n − 1 is a multiple of 6.” We will show P(n) is true for all n ∈ N. First we establish the base case, P(0). Since 70 − 1 0, and 0 is a multiple of 6, P(0) is true. Now for the inductive case. Assume P(k) holds for an arbitrary k ∈ N. That is, 7k − 1 is a multiple of 6, or in other words, 7k − 1 6j for some integer j. Now consider 7k+1 − 1: 7k+1 − 1 7k+1 − 7 + 6 7(7k − 1) + 6 by cleverness: − 1 −7 + 6 factor out a 7 from the ﬁrst two terms 7(6j) + 6 by the inductive hypothesis 6(7j + 1) factor out a 6 Therefore 7k+1 − 1 is a multiple of 6, or in other words, P(k + 1) is true. Therefore by the principle of mathematical induction, P(n) is true for all n ∈ N. qed 2.5.4. Proof. Let P(n) be the statement 1 + 3 + 5 + · · · + (2n − 1) n 2 . We will prove that P(n) is true for all n ≥ 1. First the base case, P(1). We have 1 12 which is true, so P(1) is established. Now the inductive case. Assume that P(k) is true for some ﬁxed arbitrary k ≥ 1. That is, 1 + 3 + 5 + · · · + (2k − 1) k 2 . We will now prove that P(k + 1) is also true (i.e., that 1 + 3 + 5 + · · · + (2k + 1) (k + 1)2 ). We start with the left-hand side of P(k + 1) and work to the right-hand side: 1 + 3 + 5 + · · · + (2k − 1) + (2k + 1) k 2 + (2k + 1) (k + 1)2 by ind. hyp. by factoring Thus P(k + 1) holds, so by the principle of mathematical induction, P(n) is true for all n ≥ 1. qed 2.5.5. Proof. Let P(n) be the statement F0 + F2 + F4 + · · · + F2n F2n+1 − 1. We will show that P(n) is true for all n ≥ 0. First the base case is easy because F0 0 and F1 1 so F0 F1 −1. Now consider the inductive case. Assume P(k) is true, that is, assume F0 + F2 + F4 + · · · + F2k F2k+1 − 1. To establish P(k + 1) we work from left to right: F0 + F2 + · · · + F2k + F2k+2 F2k+1 − 1 + F2k+2 by ind. hyp. F2k+1 + F2k+2 − 1 F2k+3 − 1 by recursive def. Therefore F0 + F2 + F4 + · · · + F2k+2 F2k+3 − 1, which is to say P(k + 1) holds. Therefore by the principle of mathematical induction, P(n) is true for all n ≥ 0. qed 364 B. Selected Solutions 2.5.6. Proof. Let P(n) be the statement 2n < n!. We will show P(n) is true for all n ≥ 4. First, we check the base case and see that yes, 24 < 4! (as 16 < 24) so P(4) is true. Now for the inductive case. Assume P(k) is true for an arbitrary k ≥ 4. That is, 2k < k!. Now consider P(k + 1): 2k+1 < (k + 1)!. To prove this, we start with the left side and work to the right side. 2k+1 2 · 2k < 2 · k! < (k + 1) · k! by the inductive hypothesis since k + 1 > 2 (k + 1)! Therefore 2k+1 < (k + 1)! so we have established P(k + 1). Thus by the principle of mathematical induction P(n) is true for all n ≥ 4. qed 2.5.12. The only problem is that we never established the base case. Of course, when n 0, 0 + 3 , 0 + 7. 2.5.13. Proof. Let P(n) be the statement that n + 3 < n + 7. We will prove that P(n) is true for all n ∈ N. First, note that the base case holds: 0 + 3 < 0 + 7. Now assume for induction that P(k) is true. That is, k + 3 < k + 7. We must show that P(k + 1) is true. Now since k + 3 < k + 7, add 1 to both sides. This gives k + 3 + 1 < k + 7 + 1. Regrouping (k + 1) + 3 < (k + 1) + 7. But this is simply P(k + 1). Thus by the principle of mathematical induction P(n) is true for all n ∈ N. qed 2.5.14. The problem here is that while P(0) is true, and while P(k) → P(k + 1) for some values of k, there is at least one value of k (namely k 99) when that implication fails. For a valid proof by induction, P(k) → P(k+1) must be true for all values of k greater than or equal to the base case. 2.5.16. We once again failed to establish the base case: when n 0, n 2 + n 0 which is even, not odd. 2.5.19. The proof will by by strong induction. Proof. Let P(n) be the statement “n is either a power of 2 or can be written as the sum of distinct powers of 2.” We will show that P(n) is true for all n ≥ 1. Base case: 1 20 is a power of 2, so P(1) is true. Inductive case: Suppose P(k) is true for all k < n. Now if n is a power of 2, we are done. If not, let 2x be the largest power of 2 strictly less than n. Consider n − 2x , which is a smaller number, in fact smaller than both n and 2x . Thus n − 2x is either a power of 2 or can be written as the sum of Selected Solutions 365 distinct powers of 2, but none of them are going to be 2x , so the together with 2x we have written n as the sum of distinct powers of 2. Therefore, by the principle of (strong) mathematical induction, P(n) is true for all n ≥ 1. qed 2.5.25. The idea here is that if we take the logarithm of a n , we can increase n by 1 if we multiply by another a (inside the logarithm). This results in adding 1 more log(a) to the total. Proof. Let P(n) be the statement log(a n ) n log(a). The base case, P(2) is true, because log(a 2 ) log(a · a) log(a) + log(a) 2 log(a), by the product rule for logarithms. Now assume, for induction, that P(k) is true. That is, log(a k ) k log(a). Consider log(a k+1 ). We have log(a k+1 ) log(a k · a) log(a k ) + log(a) k log(a) + log(a), with the last equality due to the inductive hypothesis. But this simpliﬁes to (k + 1) log(a), establishing P(k + 1). Therefore by the principle of mathematical induction, P(n) is true for all n ≥ 2. qed 2.6 Chapter Review 2.6.1. 430·107 2 23005. 2.6.2. (a) n + 2 terms. (b) 4n + 2. (c) (4n+8)(n+2) . 2 2.6.3. (a) 2, 10, 50, 250, . . . The sequence is geometric. (b) 2−2·525 −4 . 2.6.5. a n n 2 + 4n − 1. 2.6.6. (a) The sequence of partial sums will be a degree 4 polynomial (its sequence of diﬀerences will be the original sequence). (b) The sequence of second diﬀerences will be a degree 1 polynomial an arithmetic sequence. 2.6.7. (a) 4, 6, 10, 16, 26, 42, . . .. (b) No, taking diﬀerences gives the original sequence back, so the differences will never be constant. 366 B. Selected Solutions 2.6.8. b n (n + 3)n. 2.6.10. (a) 1, 2, 16, 68, 364, . . . . (b) a n 37 (−2)n + 74 5n . 2.6.11. (a) a2 14. a 3 52. (b) a n 16 (−2)n + 65 4n . 2.6.12. (a) On the ﬁrst day, your 2 mini bunnies become 2 large bunnies. On day 2, your two large bunnies produce 4 mini bunnies. On day 3, you have 4 mini bunnies (produced by your 2 large bunnies) plus 6 large bunnies (your original 2 plus the 4 newly matured bunnies). On day 4, you will have 12 mini bunnies (2 for each of the 6 large bunnies) plus 10 large bunnies (your previous 6 plus the 4 newly matured). The sequence of total bunnies is 2, 2, 6, 10, 22, 42 . . . starting with a0 2 and a 1 2. (b) a n a n−1 + 2a n−2 . This is because the number of bunnies is equal to the number of bunnies you had the previous day (both mini and large) plus 2 times the number you had the day before that (since all bunnies you had 2 days ago are now large and producing 2 new bunnies each). (c) Using the characteristic root technique, we ﬁnd a n a2n + b(−1)n , and we can ﬁnd a and b to give a n 34 2n + 32 (−1)n . 2.6.17. Let P(n) be the statement, “every set containing n elements has 2n diﬀerent subsets.” We will show P(n) is true for all n ≥ 1. Base case: Any set with 1 element {a} has exactly 2 subsets: the empty set and the set itself. Thus the number of subsets is 2 21 . Thus P(1) is true. Inductive case: Suppose P(k) is true for some arbitrary k ≥ 1. Thus every set containing exactly k elements has 2k diﬀerent subsets. Now consider a set containing k + 1 elements: A {a 1 , a2 , . . . , a k , a k+1 }. Any subset of A must either contain a k+1 or not. In other words, a subset of A is just a subset of {a 1 , a 2 , . . . , a k } with or without a k+1 . Thus there are 2k subsets of A which contain a k+1 and another 2k+1 subsets of A which do not contain a k+1 . This gives a total of 2k + 2k 2 · 2k 2k+1 subsets of A. But our choice of A was arbitrary, so this works for any subset containing k + 1 elements, so P(k + 1) is true. Therefore, by the principle of mathematical induction, P(n) is true for all n ≥ 1. Selected Solutions 367 3.1 Exercises 3.1.1. (a) P: it’s your birthday; Q: there will be cake. (P ∨ Q) → Q (b) Hint: you should get three T’s and one F. (c) Only that there will be cake. (d) It’s NOT your birthday! (e) It’s your birthday, but the cake is a lie. 3.1.2. 3.1.3. P T T F F (P ∨ Q) → (P ∧ Q) T F F T Q T F T F P T T F F Q T F T F ¬P ∧ (Q → P) F F F T If the statement is true, then both P and Q are false. 3.1.6. Make a truth table for each and compare. The statements are logically equivalent. 3.1.8. (a) P ∧ Q. (b) (¬P ∨ ¬R) → (Q ∨ ¬R) or, replacing the implication with a disjunction ﬁrst: (P ∧ Q) ∨ (Q ∨ ¬R). (c) (P ∧ Q) ∧ (R ∧ ¬R). This is necessarily false, so it is also equivalent to P ∧ ¬P. (d) Either Sam is a woman and Chris is a man, or Chris is a woman. 3.1.12. The deduction rule is valid. To see this, make a truth table which contains P ∨ Q and ¬P (and P and Q of course). Look at the truth value of Q in each of the rows that have P ∨ Q and ¬P true. 3.1.16. (a) ∀x∃y(O(x) ∧ ¬E(y)). (b) ∃x∀y(x ≥ y ∨ ∀z(x ≥ z ∧ y ≥ z)). 368 B. Selected Solutions (c) There is a number n for which every other number is strictly greater than n. (d) There is a number n which is not between any other two numbers. 3.2 Exercises 3.2.1. (a) For all integers a and b, if a or b is not even, then a + b is not even. (b) For all integers a and b, if a and b are even, then a + b is even. (c) There are numbers a and b such that a + b is even but a and b are not both even. (d) False. For example, a 3 and b 5. a + b 8, but neither a nor b are even. (e) False, since it is equivalent to the original statement. (f) True. Let a and b be integers. Assume both are even. Then a 2k and b 2j for some integers k and j. But then a+b 2k+2 j 2(k+ j) which is even. (g) True, since the statement is false. 3.2.2. (a) Proof by contradiction. Start of proof: Assume, for the sake of contradiction, that there are integers x and y such that x is a prime greater than 5 and x 6y + 3. End of proof: . . . this is a contradiction, so there are no such integers. (b) Direct proof. Start of proof: Let n be an integer. Assume n is a multiple of 3. End of proof: Therefore n can be written as the sum of consecutive integers. (c) Proof by contrapositive. Start of proof: Let a and b be integers. Assume that a and b are even. End of proof: Therefore a 2 + b 2 is even. 3.2.3. (a) Direct proof. Proof. Let n be an integer. Assume n is even. Then n 2k for some integer k. Thus 8n 16k 2(8k). Therefore 8n is even. qed (b) The converse is false. That is, there is an integer n such that 8n is even but n is odd. For example, consider n 3. Then 8n 24 which is even but n 3 is odd. Selected Solutions 369 3.2.4. (a) This is an example of the pigeonhole principle. We can prove it by contrapositive. Proof. Suppose that each number only came up 6 or fewer times. So there are at most six 1’s, six 2’s, and so on. That’s a total of 36 dice, so you must not have rolled all 40 dice. qed (b) We can have 9 dice without any four matching or any four being all diﬀerent: three 1’s, three 2’s, three 3’s. We will prove that whenever you roll 10 dice, you will always get four matching or all being diﬀerent. Proof. Suppose you roll 10 dice, but that there are NOT four matching rolls. This means at most, there are three of any given value. If we only had three diﬀerent values, that would be only 9 dice, so there must be 4 diﬀerent values, giving 4 dice that are all diﬀerent. qed 370 B. Selected Solutions 3.3 Chapter Review 3.3.1. P T T T T F F F F Q T T F F T T F F R T F T F T F T F ¬P → (Q ∧ R) T T T T T F F F 3.3.2. Peter is not tall and Robert is not skinny. You must be in row 6 in the truth table above. 3.3.3. Yes. To see this, make a truth table for each statement and compare. 3.3.4. Make a truth table that includes all three statements in the argument: P Q R P → Q P → R P → (Q ∧ R) T T T T T T T T F T F F T F T F T F T F F F F F F T T T T T F T F T T T F F T T T T F F F T T T Notice that in every row for which both P → Q and P → R is true, so is P → (Q ∧ R). Therefore, whenever the premises of the argument are true, so is the conclusion. In other words, the deduction rule is valid. 3.3.5. (a) Negation: The power goes oﬀ and the food does not spoil. Converse: If the food spoils, then the power went oﬀ. Contrapositive: If the food does not spoil, then the power did not go oﬀ. (b) Negation: The door is closed and the light is on. Converse: If the light is oﬀ then the door is closed. Contrapositive: If the light is on then the door is open. Selected Solutions 371 (c) Negation: ∃x(x < 1 ∧ x 2 ≥ 1) Converse: ∀x(x 2 < 1 → x < 1) Contrapositive: ∀x(x 2 ≥ 1 → x ≥ 1). (d) Negation: There is a natural number n which is prime but not solitary. Converse: For all natural numbers n, if n is solitary, then n is prime. Contrapositive: For all natural numbers n, if n is not solitary then n is not prime. (e) Negation: There is a function which is diﬀerentiable and not continuous. Converse: For all functions f , if f is continuous then f is diﬀerentiable. Contrapositive: For all functions f , if f is not continuous then f is not diﬀerentiable. (f) Negation: There are integers a and b for which a · b is even but a or b is odd. Converse: For all integers a and b, if a and b are even then ab is even. Contrapositive: For all integers a and b, if a or b is odd, then ab is odd. (g) Negation: There are integers x and y such that for every integer n, x > 0 and nx ≤ y. Converse: For every integer x and every integer y there is an integer n such that if nx > y then x > 0. Contrapositive: For every integer x and every integer y there is an integer n such that if nx ≤ y then x ≤ 0. (h) Negation: There are real numbers x and y such that x y 0 but x , 0 and y , 0. Converse: For all real numbers x and y, if x 0 or y 0 then x y 0 Contrapositive: For all real numbers x and y, if x , 0 and y , 0 then x y , 0. (i) Negation: There is at least one student in Math 228 who does not understand implications but will still pass the exam. Converse: For every student in Math 228, if they fail the exam, then they did not understand implications. Contrapositive: For every student in Math 228, if they pass the exam, then they understood implications. 372 B. Selected Solutions 3.3.6. (a) The statement is true. If n is an even integer less than or equal to 7, then the only way it could not be negative is if n was equal to 0, 2, 4, or 6. (b) There is an integer n such that n is even and n ≤ 7 but n is not negative and n < {0, 2, 4, 6}. This is false, since the original statement is true. (c) For all integers n, if n is not negative and n < {0, 2, 4, 6} then n is odd or n > 7. This is true, since the contrapositive is equivalent to the original statement (which is true). (d) For all integers n, if n is negative or n ∈ {0, 2, 4, 6} then n is even and n ≤ 7. This is false. n −3 is a counterexample. 3.3.7. (a) For any number x, if it is the case that adding any number to x gives that number back, then multiplying any number by x will give 0. This is true (of the integers or the reals). The “if” part only holds if x 0, and in that case, anything times x will be 0. (b) The converse in words is this: for any number x, if everything times x is zero, then everything added to x gives itself. Or in symbols: ∀x(∀z(x · z 0) → ∀y(x + y y)). The converse is true: the only number which when multiplied by any other number gives 0 is x 0. And if x 0, then x + y y. (c) The contrapositive in words is: for any number x, if there is some number which when multiplied by x does not give zero, then there is some number which when added to x does not give that number. In symbols: ∀x(∃z(x · z , 0) → ∃y(x + y , y)). We know the contrapositive must be true because the original implication is true. (d) The negation: there is a number x such that any number added to x gives the number back again, but there is a number you can multiply x by and not get 0. In symbols: ∃x(∀y(x + y y) ∧ ∃z(x · z , 0)). Of course since the original implication is true, the negation is false. 3.3.8. (a) (¬P ∨ Q) ∧ (¬R ∨ (P ∧ ¬R)). (b) ∀x∀y∀z(z x + y ∧ ∀w(x − y , w)). 3.3.9. (a) Direct proof. Selected Solutions 373 Proof. Let n be an integer. Assume n is odd. So n 2k + 1 for some integer k. Then 7n 7(2k + 1) 14k + 7 2(7k + 3) + 1. Since 7k + 3 is an integer, we see that 7n is odd. qed (b) The converse is: for all integers n, if 7n is odd, then n is odd. We will prove this by contrapositive. Proof. Let n be an integer. Assume n is not odd. Then n 2k for some integer k. So 7n 14k 2(7k) which is to say 7n is even. Therefore 7n is not odd. qed 3.3.10. (a) Suppose you only had 5 coins of each denomination. This means you have 5 pennies, 5 nickels, 5 dimes and 5 quarters. This is a total of 20 coins. But you have more than 20 coins, so you must have more than 5 of at least one type. (b) Suppose you have 22 coins, including 2k nickels, 2j dimes, and 2l quarters (so an even number of each of these three types of coins). The number of pennies you have will then be 22 − 2k − 2 j − 2l 2(11 − k − j − l). But this says that the number of pennies is also even (it is 2 times an integer). Thus we have established the contrapositive of the statement, “If you have an odd number of pennies then you have an odd number of at least one other coin type.” (c) You need 10 coins. You could have 3 pennies, 3 nickels, and 3 dimes. The 10th coin must either be a quarter, giving you 4 coins that are all diﬀerent, or else a 4th penny, nickel or dime. To prove this, assume you don’t have 4 coins that are all the same or all diﬀerent. In particular, this says that you only have 3 coin types, and each of those types can only contain 3 coins, for a total of 9 coins, which is less than 10. 4.1 Exercises 4.1.1. This is asking for the number of edges in K 10 . Each vertex (person) has degree (shook hands with) 9 (people). So the sum of the degrees is 90. However, the degrees count each edge (handshake) twice, so there are 45 edges in the graph. That is how many handshakes took place. 374 B. Selected Solutions 4.1.2. It is possible for everyone to be friends with exactly 2 people. You could arrange the 5 people in a circle and say that everyone is friends with the two people on either side of them (so you get the graph C5 ). However, it is not possible for everyone to be friends with 3 people. That would lead to a graph with an odd number of odd degree vertices which is impossible since the sum of the degrees must be even. 4.1.4. The graphs are not equal. For example, graph 1 has an edge {a, b} but graph 2 does not have that edge. They are isomorphic. One possible isomorphism is f : G1 → G2 deﬁned by f (a) d, f (b) c, f (c) e, f (d) b, f (e) a. 4.1.9. (a) For example: (b) This is not possible if we require the graphs to be connected. If not, we could take C 8 as one graph and two copies of C 4 as the other. (c) Not possible. If you have a graph with 5 vertices all of degree 4, then every vertex must be adjacent to every other vertex. This is the graph K 5 . (d) This is not possible. In fact, there is not even one graph with this property (such a graph would have 5 · 3/2 7.5 edges). 4.1.10. (a) False. (b) True. (c) True. (d) False. 4.2 Exercises 4.2.1. (a) This is not a tree since it contains a cycle. Note also that there are too many edges to be a tree, since we know that all trees with v vertices have v − 1 edges. (b) This is a tree since it is connected and contains no cycles (which you can see by drawing the graph). All paths are trees. (c) This is a tree since it is connected and contains no cycles (draw the graph). All stars are trees. (d) This is a not a tree since it is not connected. Note that there are not enough edges to be a tree. Selected Solutions 375 4.2.2. (a) This must be the degree sequence for a tree. This is because the vertex of degree 4 must be adjacent to the four vertices of degree 1 (there are no other vertices for it to be adjacent to), and thus we get a star. (b) This cannot be a tree. Each degree 3 vertex is adjacent to all but one of the vertices in the graph. Thus each must be adjacent to one of the degree 1 vertices (and not the other). That means both degree 3 vertices are adjacent to the degree 2 vertex, and to each other, so that means there is a cycle. Alternatively, count how many edges there are! (c) This might or might not be a tree. The length 4 path has this degree sequence (this is a tree), but so does the union of a 3-cycle and a length 1 path (which is not connected, so not a tree). (d) This cannot be a tree. The sum of the degrees is 28, so there are 14 edges. But there are 14 vertices as well, so we don’t have v e + 1, meaning this cannot be a tree. 4.2.6. Yes. We will prove the contrapositive. Assume G does not contain a cycle. Then G is a tree, so would have v e + 1, contrary to stipulation. 4.2.12. (a) No, although there are graphs for which this is true. For example, K 4 has a spanning tree that is a path (of three edges) and also a spanning tree that is a star (with center vertex of degree 3). (b) Yes. For a ﬁxed graph, we have a ﬁxed number v of vertices. Any spanning tree of the graph will also have v vertices, and since it is a tree, must have v − 1 edges. (c) No, although there are graph for which this is true (note that if all spanning trees are isomorphic, then all spanning trees will have the same number of leaves). Again, K 4 is a counterexample. One spanning tree is a path, with only two leaves, another spanning tree is a star with 3 leaves. 4.3 Exercises 4.3.1. No. A (connected) planar graph must satisfy Euler’s formula: v − e + f 2. Here v − e + f 6 − 10 + 5 1. . It could be planar, and 4.3.2. G has 10 edges, since 10 2+2+3+4+4+5 2 then it would have 6 faces, using Euler’s formula: 6 − 10 + f 2 means f 6. To make sure that it is actually planar though, we would need to 376 B. Selected Solutions draw a graph with those vertex degrees without edges crossing. This can be done by trial and error (and is possible). 4.3.6. Say the last polyhedron has n edges, and also n vertices. The total number of edges the polyhedron has then is (7 · 3 + 4 · 4 + n)/2 (37 + n)/2. In particular, we know the last face must have an odd number of edges. We also have that v 11. By Euler’s formula, we have 11 − (37 + n)/2 + 12 2, and solving for n we get n 5, so the last face is a pentagon. 4.3.8. Proof. Let P(n) be the statement, “every connected planar graph containing n edges satisﬁes v − n + f 2.” We will show P(n) is true for all n ≥ 0. Base case: there is only one graph with zero edges, namely a single isolated vertex. In this case v 1, f 1 and e 0, so Euler’s formula holds. Inductive case: Suppose P(k) is true for some arbitrary k ≥ 0. Now consider an arbitrary graph containing k + 1 edges (and v vertices and f faces). No matter what this graph looks like, we can remove a single edge to get a graph with k edges which we can apply the inductive hypothesis to. There are two cases: either the graph contains a cycle or it does not. If the graph contains a cycle, then pick an edge that is part of this cycle, and remove it. This will not disconnect the graph, and will decrease the number of faces by 1 (since the edge was bordering two distinct faces). So by the inductive hypothesis we will have v − k + f − 1 2. Adding the edge back will give v − (k + 1) + f 2 as needed. If the graph does not contain a cycle, then it is a tree, so has a vertex of degree 1. Then we can pick the edge to remove to be incident to such a degree 1 vertex. In this case, also remove that vertex. The smaller graph will now satisfy v − 1 − k + f 2 by the induction hypothesis (removing the edge and vertex did not reduce the number of faces). Adding the edge and vertex back gives v − (k + 1) + f 2, as required. Therefore, by the principle of mathematical induction, Euler’s formula holds for all planar graphs. qed 4.3.12. Proof. We know in any planar graph the number of faces f satisﬁes 3 f ≤ 2e since each face is bounded by at least three edges, but each edge borders two faces. Combine this with Euler’s formula: v−e+ f 2 2e ≥2 3 3v − e ≥ 6 v−e+ Selected Solutions 377 3v − 6 ≥ e. qed 4.4 Exercises 4.4.1. 2, since the graph is bipartite. One color for the top set of vertices, another color for the bottom set of vertices. 4.4.2. For example, K 6 . If the chromatic number is 6, then the graph is not planar; the 4-color theorem states that all planar graphs can be colored with 4 or fewer colors. 4.4.3. The chromatic numbers are 2, 3, 4, 5, and 3 respectively from left to right. 4.4.5. The cube can be represented as a planar graph and colored with two colors as follows: R B B B R R B R Since it would be impossible to color the vertices with a single color, we see that the cube has chromatic number 2 (it is bipartite). 4.4.9. The wheel graph below has this property. The outside of the wheel forms an odd cycle, so requires 3 colors, the center of the wheel must be diﬀerent than all the outside vertices. 4.4.12. If we drew a graph with each letter representing a vertex, and each edge connecting two letters that were consecutive in the alphabet, we would have a graph containing two vertices of degree 1 (A and Z) and the remaining 24 vertices all of degree 2 (for example, D would be adjacent to both C and E). By Brooks’ theorem, this graph has chromatic number at most 2, as that is the maximal degree in the graph and the graph is not a complete graph or odd cycle. Thus only two boxes are needed. 4.4.13. 4.5 Exercises 4.5.1. This is a question about ﬁnding Euler paths. Draw a graph with a vertex in each state, and connect vertices if their states share a border. Exactly two vertices will have odd degree: the vertices for Nevada and Utah. Thus you must start your road trip at in one of those states and end it in the other. 378 B. Selected Solutions 4.5.2. (a) K 4 does not have an Euler path or circuit. (b) K 5 has an Euler circuit (so also an Euler path). (c) K 5,7 does not have an Euler path or circuit. (d) K 2,7 has an Euler path but not an Euler circuit. (e) C 7 has an Euler circuit (it is a circuit graph!) (f) P7 has an Euler path but no Euler circuit. 4.5.8. If we build one bridge, we can have an Euler path. Two bridges must be built for an Euler circuit. 4.6 Exercises 4.6.1. The ﬁrst and third graphs have a matching, shown in bold (there are other matchings as well). The middle graph does not have a matching. If you look at the three circled vertices, you see that they only have two neighbors, which violates the matching condition |N(S)| ≥ |S| (the three circled vertices form the set S). 4.7 Chapter Review 4.7.1. The ﬁrst and the third graphs are the same (try dragging vertices around to make the pictures match up), but the middle graph is diﬀerent (which you can see, for example, by noting that the middle graph has only one vertex of degree 2, while the others have two such vertices). 4.7.2. The ﬁrst (and third) graphs contain an Euler path. All the graphs are planar. 4.7.3. For example, K 5 . 4.7.4. For example, K 3,3 . Selected Solutions 379 4.7.5. (a) Yes, the graphs are isomorphic, which you can see by drawing them. One isomorphism is: a b c d e f u z v x w f y g . t (b) This is easy to do if you draw the picture. Here is such a graph: Any labeling of this graph will be not isomorphic to G. For example, we could take V ′′ {a, b, c, d, e, f , g} and E′′ {ab, ac, ad, be , c f , dg}. (c) The degree sequence for G is (3, 3, 2, 1, 1, 1, 1). (d) In general this should be possible: the degree sequence does not determine the graph’s isomorphism class. However, in this case, I was almost certain this was not possible. That is, until I stumbled up this: (e) G is a tree (there are no cycles) and as such also bipartite. (f) Yes, all trees are planar. You can draw them in the plane without edges crossing. (g) The chromatic number of G is 2. It shouldn’t be hard to give a 2coloring (for example, color a, d, e , g red and b, c, f blue), but we know that all bipartite graphs have chromatic number 2. (h) It is clear from the drawing that there is no Euler path, let alone an Euler circuit. Also, since there are more than 2 vertices of odd degree, we know for sure there is no Euler path. 4.7.6. Yes. According to Euler’s formula it would have 2 faces. It does. The only such graph is C 10 . 4.7.7. (a) Only if n ≥ 6 and is even. (b) None. 380 B. Selected Solutions (c) 12. Such a graph would have 5n 2 edges. If the graph is planar, then 5n n − 2 + f 2 so there would be 4+3n faces. Also, we must 2 have 3 f ≤ 2e, since the graph is simple. So we must have 3 4+3n ≤ 5n. 2 Solving for n gives n ≥ 12. 4.7.8. (a) There were 24 couples: 6 choices for the girl and 4 choices for the boy. (b) There were 45 couples: people to dance together. 10 2 since we must choose two of the 10 (c) For part (a), we are counting the number of edges in K4,6 . In part (b) we count the edges of K 10 . 4.7.9. Yes, as long as n is even. If n were odd, then corresponding graph would have an odd number of odd degree vertices, which is impossible. 4.7.10. (a) No. The 9 triangles each contribute 3 edges, and the 6 pentagons contribute 5 edges. This gives a total of 57, which is exactly twice the number of edges, since each edge borders exactly 2 faces. But 57 is odd, so this is impossible. (b) Now adding up all the edges of all the 16 polygons gives a total of 64, meaning there would be 32 edges in the polyhedron. We can then use Euler’s formula v − e + f 2 to deduce that there must be 18 vertices. (c) If you add up all the vertices from each polygon separately, we get a total of 64. This is not divisible by 3, so it cannot be that each vertex belongs to exactly 3 faces. Could they all belong to 4 faces? That would mean there were 64/4 16 vertices, but we know from Euler’s formula that there must be 18 vertices. We can write 64 3x + 4y and solve for x and y (as integers). We get that there must be 10 vertices with degree 4 and 8 with degree 3. (Note the number of faces joined at a vertex is equal to its degree in graph theoretic terms.) 4.7.11. No. Every polyhedron can be represented as a planar graph, and the Four Color Theorem says that every planar graph has chromatic number at most 4. 4.7.12. K n,n has n 2 edges. The graph will have an Euler circuit when n is even. The graph will be planar only when n < 3. Selected Solutions 381 4.7.13. G has 8 edges (since the sum of the degrees is 16). If G is planar, then it will have 4 faces (since 6 − 8 + 4 2). G does not have an Euler path since there are more than 2 vertices of odd degree. 4.7.14. 7 colors. Thus K7 is not planar (by the contrapositive of the Four Color Theorem). 4.7.15. The chromatic number of K 3,4 is 2, since the graph is bipartite. You cannot say whether the graph is planar based on this coloring (the converse of the Four Color Theorem is not true). In fact, the graph is not planar, since it contains K 3,3 as a subgraph. 4.7.16. We have that K 3,4 has 7 vertices and 12 edges (each vertex in the group of 3 has degree 4). Then by Euler’s formula we have that 7 − 12 + f 2 so if the graph were planar, it would have f 7 faces. However, since the girth of the graph is 4 (there are no cycles of length 3) we get that 4 f ≤ 2e. But this would mean that 28 ≤ 24, a contradiction. 4.7.17. For all these questions, we are really coloring the vertices of a graph. You get the graph by ﬁrst drawing a planar representation of the polyhedron and then taking its planar dual: put a vertex in the center of each face (including the outside) and connect two vertices if their faces share an edge. (a) Since the planar dual of a dodecahedron contains a 5-wheel, it’s chromatic number is at least 4. Alternatively, suppose you could color the faces using 3 colors without any two adjacent faces colored the same. Take any face and color it blue. The 5 pentagons bordering this blue pentagon cannot be colored blue. Color the ﬁrst one red. Its two neighbors (adjacent to the blue pentagon) get colored green. The remaining 2 cannot be blue or green, but also cannot both be red since they are adjacent to each other. Thus a 4th color is needed. (b) The planar dual of the dodecahedron is itself a planar graph. Thus by the 4-color theorem, it can be colored using only 4 colors without two adjacent vertices (corresponding to the faces of the polyhedron) being colored identically. (c) The cube can be properly 3-colored. Color the “top” and “bottom” red, the “front” and “back” blue, and the “left” and “right” green. 4.7.18. (a) False. To prove this, we can give an example of a pair of graphs with the same chromatic number that are not isomorphic. For example, K 3,3 and K 3,4 both have chromatic number 2, but are not isomorphic. (b) False. The previous example does not work, but you can easily draw two trees that have the same number of vertices and edges but are 382 B. Selected Solutions not isomorphic. Since all trees have chromatic number 2, this is a counterexample. (c) True. If there is an isomorphism from G1 to G2 , then we have a bĳection that tells us how to match up vertices between the graph. Any proper vertex coloring of G1 will tell us how to properly color G2 , simply by coloring f (v i ) the same color as v i , for each vertex v i ∈ V. That is, color the vertices in G2 exactly how you color the corresponding vertices in G1 . Similarly, any proper vertex coloring of G2 corresponds to a proper vertex coloring of G1 . Thus the smallest number of colors needed to properly color G1 cannot be smaller than the smallest number of colors needed to properly color G2 , and vice-versa, so the chromatic numbers must be equal. 4.7.19. G has 13 edges, since we need 7 − e + 8 2. 4.7.20. (a) The graph does have an Euler path, but not an Euler circuit. There are exactly two vertices with odd degree. The path starts at one and ends at the other. (b) The graph is planar. Even though as it is drawn edges cross, it is easy to redraw it without edges crossing. (c) The graph is not bipartite (there is an odd cycle), nor complete. (d) The chromatic number of the graph is 3. 4.7.21. (a) False. For example, K 3,3 is not planar. (b) True. The graph is bipartite so it is possible to divide the vertices into two groups with no edges between vertices in the same group. Thus we can color all the vertices of one group red and the other group blue. (c) False. K 3,3 has 6 vertices with degree 3, so contains no Euler path. (d) False. K 3,3 again. (e) False. The sum of the degrees of all vertices is even for all graphs so this property does not imply that the graph is bipartite. 4.7.22. (a) If a graph has an Euler path, then it is planar. (b) If a graph does not have an Euler path, then it is not planar. (c) There is a graph which is planar and does not have an Euler path. (d) Yes. In fact, in this case it is because the original statement is false. Selected Solutions 383 (e) False. K 4 is planar but does not have an Euler path. (f) False. K 5 has an Euler path but is not planar. 5.1 Exercises 5.1.1. (a) 4 . 1−x (b) 2 . (1 − x)2 2x 3 . (1 − x)2 1 (d) . 1 − 5x 1 (e) . 1 + 3x (c) 1 . 1 − 5x 2 x . (g) (1 − x 3 )2 (f) 5.1.2. (a) 0, 4, 4, 4, 4, 4, . . .. (b) 1, 4, 16, 64, 256, . . .. (c) 0, 1, −1, 1, −1, 1, −1, . . .. (d) 0, 3, −6, 9, −12, 15, −18, . . .. (e) 1, 3, 6, 9, 12, 15, . . .. 5.1.4. Call the generating function A. Compute A−xA 4+x +2x 2 +3x 3 + x 4 x . Solving for A gives . 4x 4 + · · ·. Thus A − xA 4 + + 2 1 − x (1 − x)3 (1 − x) 1 + 2x . 1 − 3x + x 2 5.1.5. 5.1.6. Compute A− xA− x 2 A and the solve for A. The generating function x will be . 1 − x − x2 x 5.1.7. . (1 − x)(1 − x − x 2 ) 2 7 + . 1 − 5x 1 + 3x 5.1.8. 5.1.9. a n 3 · 4n−1 + 1. 5.1.12. (a) 1 . (1−x 2 )2 (b) 1 . (1+x)2 (c) 3x . (1−x)2 (d) 3x . (1−x)3 (partial sums). 384 B. Selected Solutions 5.1.13. (a) 0, 0, 1, 1, 2, 3, 5, 8, . . .. (b) 1, 0, 1, 0, 2, 0, 3, 0, 5, 0, 8, 0, . . .. (c) 1, 3, 18, 81, 405, . . .. (d) 1, 2, 4, 7, 12, 20, . . .. 5.1.15. x3 (1−x)2 + 1 1−x . 5.2 Exercises 5.2.1. Proof. Suppose a | b. Then b is a multiple of a, or in other words, b ak for some k. But then bc akc, and since kc is an integer, this says bc is a multiple of a. In other words, a | bc. qed 5.2.3. {. . . , −8, −4, 0, 4, 8, 12, . . .}, {. . . , −7, −3, 1, 5, 9, 13, . . .}, {. . . , −6, −2, 2, 6, 10, 14, . . .}, and {. . . , −5, −1, 3, 7, 11, 15, . . .}. 5.2.5. Proof. Assume a ≡ b (mod n) and c ≡ d (mod n). This means a b + kn and c d + jn for some integers k and j. Consider a − c. We have: a − c b + kn − (d + jn) b − d + (k − j)n. In other words, a−c is b−d more than some multiple of n, so a−c ≡ b−d (mod n). qed 5.2.6. (a) 3456 ≡ 1456 1 (mod 2). (b) 3456 9228 ≡ (−1)228 1 (mod 5). (c) 3456 9228 ≡ 2228 876 ≡ 176 1 (mod 7). (d) 3456 9228 ≡ 0228 0 (mod 9). 5.2.8. For all of these, just plug in all integers between 0 and the modulus to see which, if any, work. (a) No solutions. (b) x 2, x 5, x 8. (c) No solutions. 5.2.10. x 5 + 22k for k ∈ Z. 5.2.12. x 6 + 15k for k ∈ Z. Selected Solutions 385 5.2.14. We must solve 7x +5 ≡ 2 (mod 11). This gives x ≡ 9 (mod 11). In general, x 9 + 11k, but when you divide any such x by 11, the remainder will be 9. 5.2.15. Divide through by 2: 3x + 5y 16. Convert to a congruence, modulo 3: 5y ≡ 16 (mod 3), which reduces to 2y ≡ 1 (mod 3). So y ≡ 2 (mod 3) or y 2 + 3k. Plug this back into 3x + 5y 16 and solve for x, to get x 2 − 5k. So the general solution is x 2 − 5k and y 2 + 3k for k ∈ Z. 386 B. Selected Solutions Appendix C List of Symbols Symbol Description P, Q, R, S, . . . ∧ ∨ ¬ ∃ ∀ ∅ U 6 propositional (sentential) variables logical “and” (conjunction) 6 logical “or” (disjunction) 6 logical negation 6 existential quantiﬁer 15 universal quantiﬁer 15 the empty set 27 universal set (domain of discourse) 27 the set of natural numbers 27 the set of integers 27 the set of rational numbers 27 the set of real numbers 27 the power set of A 27 braces, to contain set elements. 27 “such that” 27 “is an element of” 27 “is a subset of” 27 “is a proper subset of” 27 set intersection 27 set union 27 Cartesian product 27 set diﬀerence 27 the complement of A 27 cardinality (size) of A 27 the Cartesian product of A and B 33 the image of A under f . 48 the inverse image of B under f . 48 the set of length n bit strings 72 the set of legth n bit strings with weight k. 72 the sequence a 0 , a 1 , a 2 , . . . 136 the nth triangular number 140 the nth Fibonacci number 145 the kth diﬀerences of a sequence 161 (Continued on next page) N Z Q R P(A) {, } : ∈ ⊆ ⊂ ∩ ∪ × \ A |A| A×B f (A) f −1 (B) Bn Bnk (a n )n∈N Tn Fn ∆k Page 387 388 C. List of Symbols Symbol Description P(n) the nth case we are trying to prove by induction the ultimate answer to life, etc. “therefore” the complete graph on n vertices the complete graph on n vertices. the complete bipartite graph of of m and n vertices. the cycle on n vertices the path on n + 1 vertices the chromatic number of G the maximum degree in G the chromatic index of G the set of neighbors of S. 42 ∴ Kn Kn K m,n Cn Pn χ(G) ∆(G) χ′(G) N(S) Page 177 178 198 239 241 241 241 241 268 271 272 284 Index ∆k -constant, 161 k-permutation, 83 modus ponens, 205 congruence, 312 conjunction, 6 connected, 239, 242 connectives, 5 and, 6 if and only if, 6 implies, 6 not, 6 or, 6 contradiction, 218 contrapositive, 10 proof by, 216 converse, 10 convex, 262 counterexample, 220 cube, 262 cycle, 242 additive principle, 58, 62 adjacent, 232, 242 ancestor, 251 argument, 197 arithmetic sequence, 148 biconditional, 6 bĳection, 47 binomial coeﬃcients, 76 binomial identity, 90 bipartite, 241, 242 bit string, 72 bow ties, 129, 354 breadth ﬁrst search, 253 Brooks’ Theorem, 272 De Morgan’s laws, 202 deduction rule, 205 degree, 242 degree sum formula, 240 depth ﬁrst search, 253 derangement, 115 descendant, 251 diﬀerence, of sets, 31 Diophantine equation, 320 direct proof, 215 disjoint, 58 disjunction, 6 divides, 308 divisibility relation, 308 Division algorithm, 310 Doctor Who, 24 dodecahedron, 264 domain, 39 double negation, 203 Canadians, 184 cardinality, 27, 30 Cartesian product, 33, 62 cases, 221 characteristic equation, 172 characteristic roots, 171, 172 child, 251 chromatic index, 272 chromatic number, 242, 268 clique, 271 closed formula, 138 for a function, 43 codomain, 39 coloring, 268 combination, 84 complement, 31 complete graph, 239, 242 composition, 55 conclusion, 197 conditional, 6 edges, 232 389 390 Index empty set, 27 Euler circuit, 242 Euler path, 242 existential quantiﬁer, 15 faces, 258 factorial, 82 Fibonacci sequence, 139, 145, 164, 167, 173 ﬁnite diﬀerences, 162 Four Color Theorem, 269 free variable, 15 function, 39 gcd, 316 geometric sequence, 150 girth, 261 Goldbach conjecture, 227 graph, 242 greatest commond divisor, 316 Hall’s Marriage Theorem, 285 Hamilton path, 279 handshake lemma, 240 Hanoi, 135 icosahedron, 264 if and only if, 6 if. . . , then. . . , 6 implication, 6 inclusive or, 6 induced subgraph, 238 induction, 177, 180 inductive hypothesis, 180 injection, 46 integers, 26, 27 intersection, 27, 31 isomorphic, 235 isomorphism, 236 isomorphism class, 237 iteration, 169 Königsberg, 231 lattice path, 73 law of logic, 209 logical equivalence, 201 logically valid, see law of logic magic chocolate bunnies, 195 matching, 283 mod, 312 modular arithmetic, 313 modulo n, 312 monochromatic, 273 multigraph, 239, 242 multiplicative principle, 59, 63 natural numbers, 27 negation, 6 neighbors, 284 NP-complete, 280 octahedron, 264 one-to-one, 46 onto, 45 parent, 251 partial sums, 151 Pascal’s triangle, 77 path, 243 perfect graph, 271 permutation, 83 Petersen graph, 266 PIE, 64 pigeonhole principle, 219 planar, 243, 258 Platonic solid, 262 polyhedron, 262 polynomial ﬁtting, 160 power set, 27, 29 predicate, 15 premises, 197 prime numbers, 187, 214 principle of inclusion/exclusion, 64 product notation, 144 proof by cases, 221 proof by contradiction, 218 proof by contrapositive, 216 Index proposition, 198 quantiﬁers exists, 15 for all, 15 subgraph, 243 induced, 238 subset, 28 summation notation, 143 surjection, 45 Ramsey theory, 274 range, 39 rationals, 27 reals, 27 recurrence relation, 138 recursive deﬁnition, 138 reference, self, see self reference root, 251 tautology, 201 telescoping, 168 tetrahedron, 264 Tower of Hanoi, 135 tree, 243 triangular numbers, 151 truth table, 199 truth value, 5 search breadth ﬁrst, 253 depth ﬁrst, 253 self reference, see reference, self sequence, 136 set, 24 set diﬀerence, 31 sibling, 252 Sigma notation, 143 stars and bars, 103 statement, 4 union, 27, 31 universal quantiﬁer, 15 valid, 197 Venn diagram, 33, 64 vertex coloring, 243, 268 vertices, 232 Vizing’s Theorem, 273 walk, 243 weight, of a string, 72 word, 58 391 392 Index Colophon This book was authored in PreTeXt. Mathematics 1 1 1 2 1 3 1 1 3 1 This gentle introduction to discrete 6 is4 written 1 mathematics 4 1 for first and second year math majors, especially those who intend to teach. The text began as a set of lecture notes for the 5 10 10 5 1 Colorado. This course serves discrete mathematics course at1 the University of Northern both as an introduction to topics in discrete math and as the “introduction to proof” course 6 15 20 15 6 1 1 for math majors. The course is usually taught with a large amount of student inquiry, and this text is written to help1facilitate 7 this. 21 35 35 21 7 1 8 1 28 56 70 56 28 8 1 Four main topics are covered: counting, sequences, logic, and graph theory. Along the 126 126 9 including 36 84 proofs 84 36 9proofs 1 1 by induction, and way proofs are introduced, by contradiction, combinatorial proofs. The book contains over 470 exercises, including 275 with solutions 210 252 45 10 1 so10 1 challenging. and another 100 or with 45 hints.120 Exercises range210 from120 elementary to quite 1 11 55 165 330 462 462 330 165 55 11 1 While there are many fine220 has the following 792 textbooks 495 math 924 792 available, 495 220this66text12 1 12 66 discrete 1 advantages: 1 13 78 286 715 1287 1716 1716 1287 715 286 78 13 1 – It is written to be used in an inquiry rich course. 1 14 91 364 1001 2002 3003 3432 3003 2002 1001 364 91 14 1 – It is written to be used in a course for future math teachers. 105 455 1365 3003 5005 6435 6435 5005 3003 1365 455 105 15 15open 1– It is 1 source, with low cost print editions and free electronic editions. 1 16 120 560 1820 4368 8008 11440 12870 11440 8008 4368 1820 560 120 16 1 To download the current version, or for information on obtaining the PreTeXt source, visit: http://discrete.openmathbooks.org/.