Introduction To Proof And Problem Solving

Book of Proof Third Edition

Richard Hammack


Published by Richard Hammack Richmond, Virginia

Book of Proof

Edition 3.2

© 2018 by Richard Hammack

This work is licensed under the Creative CommonsAttribution-NonCommercial-NoDerivative 4.0 International License

Typeset in 11pt TEX Gyre Schola using PDFLATEX

Cover by R. Hammack. The cover diagrams are based on a geometric construction that renders a correct perspective view of an object (here an octagonal column) from its floor plan. The method was invented by Piero della Francesca 1415–1492, a Renaissance painter and mathematician.



To my students




Preface vii

Introduction viii

I Fundamentals

1. Sets 3 1.1. Introduction to Sets 3 1.2. The Cartesian Product 8 1.3. Subsets 12 1.4. Power Sets 15 1.5. Union, Intersection, Difference 18 1.6. Complement 20 1.7. Venn Diagrams 22 1.8. Indexed Sets 25 1.9. Sets That Are Number Systems 30 1.10. Russell’s Paradox 32

2. Logic 34 2.1. Statements 35 2.2. And, Or, Not 39 2.3. Conditional Statements 42 2.4. Biconditional Statements 46 2.5. Truth Tables for Statements 48 2.6. Logical Equivalence 50 2.7. Quantifiers 53 2.8. More on Conditional Statements 56 2.9. Translating English to Symbolic Logic 57 2.10. Negating Statements 59 2.11. Logical Inference 63 2.12. An Important Note 64

3. Counting 65 3.1. Lists 65 3.2. The Multiplication Principle 67 3.3. The Addition and Subtraction Principles 74 3.4. Factorials and Permutations 78 3.5. Counting Subsets 85 3.6. Pascal’s Triangle and the Binomial Theorem 90 3.7. The Inclusion-Exclusion Principle 93 3.8. Counting Multisets 96 3.9. The Division and Pigeonhole Principles 104 3.10. Combinatorial Proof 108




II How to Prove Conditional Statements

4. Direct Proof 113 4.1. Theorems 113 4.2. Definitions 115 4.3. Direct Proof 118 4.4. Using Cases 124 4.5. Treating Similar Cases 125

5. Contrapositive Proof 128 5.1. Contrapositive Proof 128 5.2. Congruence of Integers 131 5.3. Mathematical Writing 133

6. Proof by Contradiction 137 6.1. Proving Statements with Contradiction 138 6.2. Proving Conditional Statements by Contradiction 141 6.3. Combining Techniques 142 6.4. Some Words of Advice 143

III More on Proof

7. Proving Non-Conditional Statements 147 7.1. If-and-Only-If Proof 147 7.2. Equivalent Statements 149 7.3. Existence Proofs; Existence and Uniqueness Proofs 150 7.4. Constructive Versus Non-Constructive Proofs 154

8. Proofs Involving Sets 157 8.1. How to Prove a ∈ A 157 8.2. How to Prove A ⊆ B 159 8.3. How to Prove A = B 162 8.4. Examples: Perfect Numbers 165

9. Disproof 172 9.1. Counterexamples 174 9.2. Disproving Existence Statements 176 9.3. Disproof by Contradiction 178

10. Mathematical Induction 180 10.1. Proof by Induction 182 10.2. Proof by Strong Induction 187 10.3. Proof by Smallest Counterexample 191 10.4. The Fundamental Theorem of Arithmetic 192 10.5. Fibonacci Numbers 193




IV Relations, Functions and Cardinality

11. Relations 201 11.1. Relations 201 11.2. Properties of Relations 205 11.3. Equivalence Relations 210 11.4. Equivalence Classes and Partitions 215 11.5. The Integers Modulo n 218 11.6. Relations Between Sets 221

12. Functions 223 12.1. Functions 223 12.2. Injective and Surjective Functions 228 12.3. The Pigeonhole Principle Revisited 233 12.4. Composition 235 12.5. Inverse Functions 238 12.6. Image and Preimage 242

13. Proofs in Calculus 244 13.1. The Triangle Inequality 245 13.2. Definition of a Limit 246 13.3. Limits That Do Not Exist 249 13.4. Limit Laws 251 13.5. Continuity and Derivatives 256 13.6. Limits at Infinity 258 13.7. Sequences 261 13.8. Series 265

14. Cardinality of Sets 269 14.1. Sets with Equal Cardinalities 269 14.2. Countable and Uncountable Sets 275 14.3. Comparing Cardinalities 280 14.4. The Cantor-Bernstein-Schröder Theorem 284

Conclusion 291

Solutions 292



Preface to the Third Edition

My goal in writing this book has been to create a very inexpensivehigh-quality textbook. The book can be downloaded from my web page in PDF format for free, and the print version costs considerably less than comparable traditional textbooks.

In this third edition, Chapter 3 (on counting) has been expanded, and a new chapter on calculus proofs has been added. New examples and exercises have been added throughout. My decisions regarding revisions have been guided by both the Amazon reviews and emails from readers, and I am grateful for all comments.

I have taken pains to ensure that the third edition is compatible with the second. Exercises have not been reordered, although some have been edited for clarity and some new ones have been appended. (The one exception is that Chapter 3’s reorganization shifted some exercises.) The chapter sequencing is identical between editions, with one exception: The final chapter on cardinality has become Chapter 14 in order to make way for the new Chapter 13 on calculus proofs. There has been a slight renumbering of the sections within chapters 10 and 11, but the numbering of the exercises within the sections is unchanged.

This core of this book is an expansion and refinement of lecture notes I developed while teaching proofs courses over the past 18 years at Virginia Commonwealth University (a large state university) and Randolph-Macon College (a small liberal arts college). I found the needs of these two audiences to be nearly identical, and I wrote this book for them. But I am mindful of a larger audience. I believe this book is suitable for almost any undergraduate mathematics program.

Richard Hammack Lawrenceville, Virginia February 14, 2018




This is a book about how to prove theorems. Until this point in your education, mathematics has probably been

presented as a primarily computational discipline. You have learned to solve equations, compute derivatives and integrals, multiply matrices and find determinants; and you have seen how these things can answer practical questions about the real world. In this setting your primary goal in using mathematics has been to compute answers.

But there is another side of mathematics that is more theoretical than computational. Here the primary goal is to understand mathematical structures, to provemathematical statements, and even to invent or discover new mathematical theorems and theories. The mathematical techniques and procedures that you have learned and used up until now are founded on this theoretical side of mathematics. For example, in computing the area under a curve, you use the fundamental theorem of calculus. It is because this theorem is true that your answer is correct. However, in learning calculus you were probably far more concerned with how that theorem could be applied than in understanding why it is true. But how do we know it is true? How can we convince ourselves or others of its validity? Questions of this nature belong to the theoretical realm of mathematics. This book is an introduction to that realm.

This book will initiate you into an esoteric world. You will learn and apply the methods of thought that mathematicians use to verify theorems, explore mathematical truth and create new mathematical theories. This will prepare you for advanced mathematics courses, for you will be better able to understand proofs, write your own proofs and think critically and inquisitively about mathematics.

The book is organized into four parts, as outlined below.




PART I Fundamentals • Chapter 1: Sets • Chapter 2: Logic • Chapter 3: Counting Chapters 1 and 2 lay out the language and conventions used in all advanced mathematics. Sets are fundamental because every mathematical structure, object, or entity can be described as a set. Logic is fundamental because it allows us to understand the meanings of statements, to deduce facts about mathematical structures and to uncover further structures. All subsequent chapters build on these first two chapters. Chapter 3 is included partly because its topics are central to many branches of mathematics, but also because it is a source of many examples and exercises that occur throughout the book. (However, the course instructor may choose to omit Chapter 3.)

PART II Proving Conditional Statements • Chapter 4: Direct Proof • Chapter 5: Contrapositive Proof • Chapter 6: Proof by Contradiction Chapters 4 through 6 are concerned with three main techniques used for proving theorems that have the “conditional” form “If P, then Q.”

PART III More on Proof • Chapter 7: Proving Non-Conditional Statements • Chapter 8: Proofs Involving Sets • Chapter 9: Disproof • Chapter 10: Mathematical Induction These chapters deal with useful variations, embellishments and conse- quences of the proof techniques introduced in Chapters 4 through 6.

PART IV Relations, Functions and Cardinality • Chapter 11: Relations • Chapter 12: Functions • Chapter 13: Proofs in Calculus • Chapter 14: Cardinality of Sets These final chapters are mainly concerned with the idea of functions, which are central to all of mathematics. Upon mastering this material you will be ready for advanced mathematics courses such as abstract algebra, analysis, topology, combinatorics and theory of computation.

Free PDF version



x Introduction

The chapters are organized as in the following dependency tree. The left-hand column forms the core of the book; each chapter in this column uses material from all chapters above it. Chapters 3 and 13 may be omitted without loss of continuity. But the material in Chapter 3 is a great source of exercises, and the reader who omits it should ignore the later exercises that draw from it. Chapter 10, on induction, can also be omitted with no break in continuity. However, induction is a topic that most proof courses will include.

Dependency TreeChapter 1 Sets

Chapter 2 Logic

Chapter 4 Direct Proof

Chapter 5 Contrapositive Proof

Chapter 6 Proof by Contradiction

Chapter 7 Non-Conditional Proof

Chapter 8 Proofs Involving Sets

Chapter 9 Disproof

Chapter 11 Relations

Chapter 12 Functions

Chapter 14 Cardinality of Sets

Chapter 3 Counting

Chapter 10 Mathematical Induction

Chapter 13 Proofs in Calculus

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