rdw_027 (reading, discovering and writing proofs, version 0.2.7)

Reading, Discovering and Writing Proofs

Version 0.2.7

c Steven Furino

December 10, 2012

Contents

I Introduction 12

1 In the beginning 131.1 What Makes a Mathematician a Mathematician? . . . . . . . . . . . . . . . 131.2 How The Course Works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.3 Why do we reason formally? . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.4 Reading and Lecture Schedule . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.4.1 Lecture Schedule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.4.2 Reading Schedule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2 Our First Proof 192.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.2 The Language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.3 Implications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.4 Our First Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3 Discovering Proofs 273.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.2 Discovering a Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.3 Reading A Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.4 The Division Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.5 Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

II Foundations 34

4 Truth Tables 354.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.2 Truth Tables as Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.3 Truth Tables to Evaluate Logical Expressions . . . . . . . . . . . . . . . . . 374.4 Contrapositive and Converse . . . . . . . . . . . . . . . . . . . . . . . . . . 404.5 More Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414.6 Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

5 Introduction to Sets 425.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425.2 Describing a Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425.3 Set Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

5.3.1 Venn Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485.4 Comparing Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

5.4.1 Sets of Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485.4.2 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

2

Section 0.0 CONTENTS 3

5.5 Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

6 More on Sets 516.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516.2 Showing Two Sets Are Equal . . . . . . . . . . . . . . . . . . . . . . . . . . 516.3 More Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

III Proof Techniques 54

7 Quantifiers 557.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 557.2 Quantifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 557.3 The Object Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 587.4 The Construct Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 597.5 The Select Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 627.6 Sets and Quantifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 637.7 A Non-Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

8 Nested Quantifiers 668.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 668.2 Onto (Surjective) Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

8.2.1 Definition of Function . . . . . . . . . . . . . . . . . . . . . . . . . . 668.2.2 Definition of Onto (Surjective) . . . . . . . . . . . . . . . . . . . . . 678.2.3 Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 698.2.4 Discovering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 708.2.5 A Difficult Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

8.3 Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 738.3.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 738.3.2 Reading A Limit Proof . . . . . . . . . . . . . . . . . . . . . . . . . 748.3.3 Discovering a Limit Proof . . . . . . . . . . . . . . . . . . . . . . . . 768.3.4 A Harder Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

8.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

9 Practice, Practice, Practice: Quantifiers and Sets 819.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 819.2 Worked Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 819.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 849.4 Quiz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

10 Simple Induction 8710.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8710.2 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

10.2.1 Summation Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . 8710.2.2 Product Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8910.2.3 Recurrence Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

10.3 Introduction to Induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9010.4 Principle of Mathematical Induction . . . . . . . . . . . . . . . . . . . . . . 90

10.4.1 Why Does Induction Work? . . . . . . . . . . . . . . . . . . . . . . . 9110.4.2 Two Examples of Simple Induction . . . . . . . . . . . . . . . . . . . 9110.4.3 A Different Starting Point . . . . . . . . . . . . . . . . . . . . . . . . 93

4 Chapter 0 CONTENTS

10.5 An Interesting Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9510.6 Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

11 Strong Induction 9911.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9911.2 Strong Induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9911.3 Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

12 Binomial Theorem 10512.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10512.2 Binomial Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10512.3 More Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10912.4 Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

13 Negation 11013.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11013.2 Negating Statements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11013.3 Negating Statements with Quantifiers . . . . . . . . . . . . . . . . . . . . . 112

13.3.1 Counterexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11313.4 Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

14 Contradiction 11614.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11614.2 How To Use Contradiction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

14.2.1 When To Use Contradiction . . . . . . . . . . . . . . . . . . . . . . . 11714.2.2 Reading a Proof by Contradiction . . . . . . . . . . . . . . . . . . . 11714.2.3 Discovering and Writing a Proof by Contradiction . . . . . . . . . . 118

15 Contrapositive 12115.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12115.2 The Contrapositive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

15.2.1 When To Use The Contrapositive . . . . . . . . . . . . . . . . . . . . 12115.3 Reading a Proof That Uses the Contrapositive . . . . . . . . . . . . . . . . 121

15.3.1 Discovering and Writing a Proof Using The Contrapositive . . . . . 123

16 Uniqueness 12516.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12516.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12516.3 Showing X = Y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12616.4 Finding a Contradiction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12716.5 The Division Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

17 Elimination 13117.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13117.2 When to Use the Elimination Method . . . . . . . . . . . . . . . . . . . . . 13117.3 How to Use the Elimination Method . . . . . . . . . . . . . . . . . . . . . . 13117.4 Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13217.5 Writing and Discovering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133


IV Securing Internet Commerce 135

18 The Greatest Common Divisor 13618.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13618.2 Greatest Common Divisor . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13618.3 Certificate of Correctess . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14018.4 More Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14218.5 Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

19 The Extended Euclidean Algorithm 14419.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14419.2 The Extended Euclidean Algorithm (EEA) . . . . . . . . . . . . . . . . . . 14419.3 More Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

20 Properties Of GCDs 14920.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14920.2 Some Useful Propositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14920.3 Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

21 Linear Diophantine Equations:One Solution 15621.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15621.2 Linear Diophantine Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 156

21.2.1 Finding One Solution to ax+ by = c . . . . . . . . . . . . . . . . . . 157

22 Linear Diophantine Equations:All Solutions 16122.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16122.2 Finding All Solutions to ax+ by = c . . . . . . . . . . . . . . . . . . . . . . 16122.3 More Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16522.4 Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

23 Congruence 16823.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16823.2 Congruences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

23.2.1 Definition of Congruences . . . . . . . . . . . . . . . . . . . . . . . . 16823.3 Elementary Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16923.4 Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

24 Congruence and Remainders 17424.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17424.2 Congruence and Remainders . . . . . . . . . . . . . . . . . . . . . . . . . . . 17424.3 More Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17824.4 Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

25 Modular Arithmetic 17925.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17925.2 Modular Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

25.2.1 [0] Zm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18125.2.2 [1] Zm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18125.2.3 Identities and Inverses in Zm . . . . . . . . . . . . . . . . . . . . . . 18225.2.4 Subtraction in Zm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183


25.2.5 Division in Zm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

26 Fermats Little Theorem 18426.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18426.2 Fermats Little Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18426.3 Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

27 Linear Congruences 19027.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19027.2 The Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19027.3 Extending Equivalencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19327.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19327.5 More Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19527.6 Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196

28 Chinese Remainder Theorem 19728.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19728.2 An Old Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19728.3 Chinese Remainder Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 19828.4 More Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20128.5 Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

29 Practice, Practice, Practice: Congruences 20329.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20329.2 Worked Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20329.3 Quiz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21129.4 Preparing for RSA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

30 The RSA Scheme 21430.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21430.2 Private Key Cryptography . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

30.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21430.2.2 Substitution Cipher . . . . . . . . . . . . . . . . . . . . . . . . . . . 21530.2.3 Looking for Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . 21730.2.4 Vigene`re Ciphers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

30.3 Why Public Key Cryptography? . . . . . . . . . . . . . . . . . . . . . . . . 22030.4 Implementing RSA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

30.4.1 Setting up RSA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22130.4.2 Sending a Message . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22130.4.3 Receiving a Message . . . . . . . . . . . . . . . . . . . . . . . . . . . 22130.4.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222

30.5 Does M = R? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22330.6 How Secure Is RSA? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225

V Bijections and Counting 226

31 Injections and Bijections 22731.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22731.2 One-to-one (Injective) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227

31.2.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22731.2.2 Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228


31.2.3 Discovering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22931.2.4 A Difficult Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230

31.3 Bijections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23131.4 Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232

32 Counting 23332.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23332.2 African Shepherds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23332.3 What Does It Mean To Count? . . . . . . . . . . . . . . . . . . . . . . . . . 23432.4 Showing That A Bijection Exists . . . . . . . . . . . . . . . . . . . . . . . . 23432.5 Finite Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236

33 Cardinality of Infinite Sets 24033.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24033.2 Infinite Sets Are Weird . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24033.3 Infinite Sets are Even Weirder Than You Thought . . . . . . . . . . . . . . 24233.4 Not All Infinite Sets Have The Same Cardinality . . . . . . . . . . . . . . . 24333.5 More Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244

34 Practice, Practice, Practice: Bijections and Cardinality 24634.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24634.2 Worked Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24634.3 Quiz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249

VI Complex Numbers and Eulers Formula 250

35 Complex Numbers 25135.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25135.2 Different Equations Require Different Number Systems . . . . . . . . . . . . 25135.3 Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25235.4 More Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256

36 Properties Of Complex Numbers 25736.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25736.2 Conjugate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25736.3 Modulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25936.4 More Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26036.5 Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261

37 Graphical Representations of Complex Numbers 26237.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26237.2 The Complex Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262

37.2.1 Cartesian Coordinates (x, y) . . . . . . . . . . . . . . . . . . . . . . . 26237.2.2 Modulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263

37.3 Polar Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26337.4 Converting Between Representations . . . . . . . . . . . . . . . . . . . . . . 264

38 De Moivres Theorem 26738.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26738.2 De Moivres Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26738.3 Complex Exponentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269


38.4 More Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27038.5 Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270

39 Roots of Complex Numbers 27139.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27139.2 Complex n-th Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27139.3 Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274

40 Practice, Practice, Practice:Complex Numbers 27540.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27540.2 Worked Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27540.3 Quiz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278

VII Factoring Polynomials 279

41 An Introduction to Polynomials 28041.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28041.2 Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28041.3 Operations on Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . 281

42 Factoring Polynomials 28542.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28542.2 Polynomial Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285

43 Practice, Practice, Practice:Polynomials 29043.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29043.2 Worked Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29043.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29243.4 Quiz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29343.5 Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294

44 Practice, Practice, Practice: Course Review 29544.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29544.2 Suggestions On How To Start A Proof . . . . . . . . . . . . . . . . . . . . . 29644.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297

VIII Finding the Shortest Path 300

45 The Shortest Path Problem 30145.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30145.2 The Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30145.3 Abstraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30345.4 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30445.5 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304

46 Paths, Walks, Cycles and Trees 30546.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30546.2 The Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305


46.3 Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308

47 Trees 31147.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31147.2 Properties of Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311

48 Dijkstras Algorithm 31548.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31548.2 Dijkstras Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31548.3 Certificate of Optimality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320

49 Certificate of Optimality - Path 32249.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32249.2 Certificate of Optimality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32249.3 Weighted Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32349.4 Certificate of Optimality - Tree . . . . . . . . . . . . . . . . . . . . . . . . . 328

IX An Introduction to Fermats Last Theorem 330

50 Introduction to Primes 33150.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33150.2 Introduction to Primes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33150.3 Induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33250.4 Fundamental Theorem of Arithmetic . . . . . . . . . . . . . . . . . . . . . . 33350.5 Finding a Prime Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33550.6 Working With Prime Factorizations . . . . . . . . . . . . . . . . . . . . . . 33750.7 More Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33850.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339

51 Introduction to Fermats Last Theorem 34051.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34051.2 History of Fermats Last Theorem . . . . . . . . . . . . . . . . . . . . . . . 34051.3 Pythagorean Triples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342

52 Characterization of Pythagorean Triples 34652.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34652.2 Pythagorean Triples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346

53 Fermats Theorem for n = 4 34953.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34953.2 n = 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34953.3 Reducing the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35153.4 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352

54 Problems Related to FLT 35354.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35354.2 x4 y4 = z2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35354.3 Pythagorean Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355

55 Practice, Practice, Practice:


Primes and Non-Linear Diophantine Equations 35655.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35655.2 Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356

56 Appendix 358

Preface

These notes are the script for the online lectures of MATH 135 at the University of Waterlooin Fall 2012. The script has been supplemented by worked examples and exercises.

These notes are very much a work in progress. Please send any corrections or suggestionsto Steven Furino at [email protected]

11

Part I

Introduction

12

Chapter 1

In the beginning

1.1 What Makes a Mathematician a Mathematician?

Welcome to MATH 135!

Let me begin with a question. What makes a mathematician a mathematician?

Many people would answer that someone who works with numbers is a mathematician.But bookkeepers for small businesses work with numbers and we dont normally consider abookkeeper as a mathematician. Others might think of geometry and answer that someonewho works with shapes is a mathematician. But architects work with shapes and we dontnormally consider architects as mathematicians. Still others might answer that people whouse formulas are mathematicians. But engineers work with formulas and we dont normallyconsider engineers as mathematicians. A more insightful answer would be that people whofind patterns and provide descriptions and evidence for those patterns are mathematicians.But scientists search for and document patterns and we dont normally consider scientistsas mathematicians.

The answer is proof - a rigorous, formal argument that establishes the truth of a statement.This has been the defining characteristic of mathematics since ancient Greece.

This course is about reading, writing and discovering proofs. If you have never done thisbefore, do not worry. The course will provide you with techniques that will help, and wewill practice those techniques in the context of some very interesting algebra.

1.2 How The Course Works

He who seeks for methods without having a definite problem in mind seeks forthe most part in vain.

David Hilbert

Let me describe how the course works.

Throughout the course, we will work on four problems all of which illustrate the need forproof. The first problem resolves a very important practical commercial problem. Thesecond problem concerns an astonishing result about one the simplest things we do, count.The third problem results in a new number system and yields a surprising and beautiful

13

14 Chapter 1 In the beginning

formula. The fourth problem relies on a profound theorem proved by Carl Friedrich Gauss,the greatest mathematician of the modern age. Here are the four problems.

How do we secure internet commerce? Have you ever bought a song or movie overiTunes? Have you ever done your banking over the web? How do you make surethat your credit card number and personal information are not intercepted by badguys? Number theory allows us to enable secure web transactions. And that theoryis backed by proof.

What does it mean to count? You probably learned to count before you went to school.But how do you count to infinity? And is there only one infinity?

Why does eipi + 1 = 0 ? e is a very unusual number that arises in calculus. i is a veryunusual number because it has the property that i2 = 1. pi is a very unusualnumber even if it is common. It is the unique ratio of the circumference of a circleto its diameter. Why should that ratio be unique? One is the basis of the naturalnumbers, hence the integers, hence the rationals. Zero is a difficult number and wasonly accepted into the mathematics of western Europe because of the influence ofHindu and Islamic scholars. Why should all of these numbers be connected in sosimple and elegant a form?

How do we factor polynomials? You have factored integers into a product of primenumbers. There is also a need in mathematics to factor polynomials, expressions likeax4 + bx3 + cx2 + dx+ e into the polynomial equivalent of prime numbers.

The course notes contain two other problems if you would like to see the power of proofdisplayed in very different contexts.

How do we find the shortest path from one point to another? How does a telecom-munications company route your cell phone call? How does Google find the quickestroute on Google maps? How does a courier company route your package? All of thesetasks are completed using a shortest path algorithm. And how do we know we havefound the shortest path? Proof.

How many positive integer solutions are there to xn + yn = zn where n is an inte-ger greater than or equal to three? This is one of the most famous problems in thehistory of mathematics and it took over 350 years to solve. It was first conjecturedby the French mathematician Pierre de Fermat in 1637 and was only solved in 1995by Andrew Wiles.

To work with these problems we will need to learn about congruences, modular arithmetic,complex numbers and polynomials. And to work with these topics, we must learn somefoundational mathematics such as logical expressions and sets, and, most importantly, wemust learn how to recognize and use proof techniques.

Section 1.3 Why do we reason formally? 15

1.3 Why do we reason formally?

But why do we reason so formally at all? Many people believe that humans already knowenough mathematics so Why bother with proofs? There are quite a few reasons.

To prevent silliness. In solving quadratic equations with non-real roots, some of you willhave encountered the number i which has the special property that i2 = 1. Butthen,

1 = i2 = i i = 11 = 11 =

1 = 1

Clearly, something is amiss.

To understand better. How would most of us answer the question Whats a real num-ber? We would probably say that any number written as a decimal expansion is areal number and any two different expansions represent different numbers. But thenwhat about this?

Let x = 0.9 = 0.999 . . . .

Multiplying by 10 and subtracting gives

10x = 9.9 x = 0.9

9x = 9

which implies x = 1, not x = 0.9.

Or suppose we wanted to evaluate the infinite sum

1 1 + 1 1 + 1 1 + 1 1 + . . .

If we pair up the first two terms we get zero and every successive pair of terms alsogives us 0 so the sum is zero.

1 1 + 1 1 +

1 1 +

1 1 + . . .

On the other hand, if we pair up the second and third term we get 0 and all successivepairs of terms give 0 so the sum is 1.

1 1 + 1

1 + 1

1 + 1

1 + 1 + . . .

Or suppose we wanted to resolve Zenos paradox. Zeno was a famous ancient Greekphilosopher who posed the following problem. Suppose the Greek hero Achilles wasgoing to race against a tortoise and suppose, in recognition of the slowness of thetortoise, that the tortoise gets a 100m head start. By the time Achilles has run halfthe distance between he and the tortoise, the tortoise has moved ahead. And nowagain, by the time Achilles has run half the remaining distance between he and thetortoise, the tortoise has moved ahead. No matter how fast Achilles runs, the tortoisewill always be ahead! You might object that your eyes see Achilles pass the tortoise,but what is logically wrong with Zenos argument?


To make better commercial decisions. Building pipelines is expensive. And lots ofpipelines will be built in the next few decades. Pipelines will ship oil, natural gas,water and sewage. Finding the shortest route given physical constraints (mountains,rivers, lakes, cities), environmental constraints (protection of the water table, no accessthrough national or state parks), and supply chain constraints (access to concrete andsteel) is very important. How do pipeline builders prove that the route they havechosen for the pipeline is the shortest possible route given the constraints?

To discover solutions. Formal reasoning provides a set of tools that allow us to thinkrationally and carefully about problems in mathematics, computing, engineering, sci-ence, economics and any discipline in which we create models.

Poor reasoning can be very expensive. Inaccurate application of financial models ledto losses of hundreds of billions of dollars during the financial crisis of 2008.

To experience joy. Mathematics can be beautiful, just as poetry can be beautiful. Butto hear the poetry of mathematics, one must first understand the language.

Section 1.4 Reading and Lecture Schedule 17

1.4 Reading and Lecture Schedule

1.4.1 Lecture Schedule

This is a proposed lecture schedule.

Lec. Ch. Topic

1 1 In The Beginning

2 2 Our First Proof

3 3 Discovering Proofs

4 7 Quantifiers

5 8 Nested Quantifiers

6 9 Practice, Practice, Practice: Quantifiers and Sets

7 10 Simple Induction

8 11 Strong Induction

9 12 Binomial Theorem

10 18 Greatest Common Divisor

11 19 Extended Euclidean Algorithm

12 20 Properties of the GCD

13 21 Linear Diophantine Equations 1

14 22 Linear Diophantine Equations 2

15 23 Congruence

16 24 Congruence and Remainders

17 25 Modular Arithmetic

18 26 Fermats Little Theorem

19 27 Linear Congruences

20 28 Chinese Remainder Theorem

21 29 Practice, Practice, Practice: Congruences

22 30 RSA

23 31 Injections and Bijections

24 32 Counting

25 33 Cardinality of Infinite Sets

26 34 Practice, Practice, Practice: Bijections and Cardinality

27 35 Introduction to Complex Numbers

28 36 Properties of Complex Numbers

29 37 Graphical Representations of Complex Numbers

30 38 DeMoivres Theorem

31 39 Roots of Complex Numbers

32 40 Practice, Practice, Practice: Complex Numbers

33 41 An Introduction to Polynomials

34 42 Factoring Polynomials

35 43 Practice, Practice, Practice: Polynomials

36 44 Practice, Practice, Practice: Course Review


1.4.2 Reading Schedule

Since one of the goals of this course is to help you become comfortable reading mathematics,there are several short chapters for you to read. After you have completed the reading, anonline assignment will help you consolidate what you know.

Ch. Topic Before Lecture

4. Truth Tables 3. Discovering Proofs

5. Introduction to Sets 3. Discovering Proofs

13. Negation 6. PPP: Quantifiers and Sets

14. Contradiction 10. Greatest Common Divisor

15. Contrapositive 10. Greatest Common Divisor

16. Uniqueness 11. Extended Euclidean Algorithm

17. Elimination 12. Properties of the GCD

30. Private Key Cryptography (30.2) 22. The RSA Scheme

Chapter 2

Our First Proof

2.1 Objectives

The technique objectives are:

1. Define statement, hypothesis, conclusion and implication.

2. Learn how to structure the analysis of a proof.

3. Carry out the analysis of a proof.

The content objectives are:

1. Define divisibility.

2. State and prove the Transitivity of Divisibility.

2.2 The Language

Mathematics is the language of mathematicians, and a proof is a method of com-municating a mathematical truth to another person who speaks the language.(Solow, How to Read and Do Proofs)

Mathematics is an unusual language. It is extraordinarily precise. When a proof is fullyand correctly presented, there is no ambiguity and no doubt about its correctness.

However, understanding a proof requires understanding the language. This course will helpyou with the basic grammar of the language of mathematics and is applicable to all proofs.Just as in learning any new language, you will need lots of practice to become fluent.

19

20 Chapter 2 Our First Proof

One of the goals of this course is to learn proof techniques. Our broad objectives for thisgoal are simple.

1. Explain and categorize proof techniques that can be used in any proof. This coursewill teach not only how a technique works, but when it is most likely to be used andwhy it works.

2. Learn how to read a proof. This will require you to identify the techniques of the firstobjective.

3. Discover your own proofs. Knowledge of technique is essential but inadequate. Or,as we would say in the language of mathematics, technique is necessary but notsufficient. Discovering your own proof requires not only technique but also under-standing, creativity, intuition and experience. This course will help with the techniqueand experience. Understanding, creativity, and intuition come with time. Talent helpsof course.

4. Write your own proofs. Having discovered a proof, you must distill your discoveryinto mathematical prose that is targeted at a specific audience.

Hopefully, in the previous lecture, I convinced you of why we need to prove things. Nowwhat is it that mathematicians prove? Mathematicians prove statements.

Definition 2.2.1

Statement

A statement is a sentence which is either true or false.

Example 1 Here are some examples of statements.

1. 2 + 2 = 4. (A true statement.)

2. 2 + 2 = 5. (A false statement.)

3. x2 1 = 0 has two distinct real roots. (A true statement.)4. There exists an angle such that sin() > 1. (A false statement.)

Example 2 Now consider the following sentences.

1. x > 0.

2. 4ABC is congruent to 4PQR.

These are statements only if we have an appropriate value for x in the first sentence andappropriate instances of 4ABC and 4PQR in the second sentence. For example, if x isthe number 5, then the sentence 5 > 0 is a statement since the sentence is true. If x isthe number 5, then the sentence 5 > 0 is also a statement since the sentence is false.The key point is that a statement is a sentence which must be true or false. If x is theEnglish word algebra, then the sentence algebra > 0 is not a statement since the sentenceis neither true nor false. Sentences like the two above are called open sentences.

Section 2.3 Implications 21

Definition 2.2.2

Open Sentence

An open sentence is a sentence that

contains one or more variables, where each variable has values that come from a designated set called the domain of the

variable, and

where the sentence is either true or false whenever values from the respective domainsof the variables are substituted for the variables.

For example, if the domain of x is the set of real numbers, then for any real number chosenand substituted for x, the sentence x > 0 is a statement.

In this course, we will treat all open sentences as statements under the assumption that thevalues of the variables always come from a suitable domain.

2.3 Implications

Definition 2.3.1

Implication

The most common type of statement we will prove is an implication. Implications havethe form

If A is true , then B is true

where A and B are themselves statements. An implication is more commonly read as

If A, then B

orA implies B

and is written symbolically asA B

Definition 2.3.2

CompoundStatement

An implication is a compound statement, that is, it is made up of more than one state-ment.

In the statement A implies B, A is a statement which may be true or false. B is astatement which may be true or false. A implies B is also a statement which may be trueor false.

Definition 2.3.3

Hypothesis,Conclusion

The statement A is called the hypothesis. The statement B is called the conclusion.


REMARK

To prove the implication A implies B, you assume that A is true and you use thisassumption to show that B is true. Statement A is what you start with. Statement B iswhere you must end up.

To use the implication A implies B, you must first establish that A is true. After youhave established that A is true, then you can invoke B.

It is crucial that you are able to identify

1. the hypothesis

2. the conclusion

3. whether you are using or proving an implication

Here are some examples of implications.

Example 3 If x is a positive real number, then log10 x > 0.

Hypothesis: x is a positive real number.

Conclusion: log10 x > 0.

Example 4 Let f(x) = x sin(x). Then f(x) = x for some real number x with 0 x 2pi.

Hypothesis: f(x) = x sin(x).

Conclusion: f(x) = x for some real number x with 0 x 2pi.

Example 5 In plane geometry, ABC = XY Z whenever 4ABC is similar to 4XY Z.

Hypothesis: All figures are in the plane. 4ABC 4XY Z.Conclusion: ABC = XY Z.

2.4 Our First Proof

Let us read our first proof. We begin with a definition.

Definition 2.4.1

Divisibility

An integer m divides an integer n, and we write m | n, if there exists an integer k so thatn = km.

Section 2.4 Our First Proof 23

Example 6

3 | 6 since we can find an integer k, 2 in this case, so that 6 = k 3. 5 - 6 since no integer k exists so that 6 = k 5. For all integers a, a | 0 since 0 = 0 a. This is true for a = 0 as well. For all non-zero integers a, 0 - a since there is no integer k so that k 0 = a.

Some comments about definitions are in order. If mathematics is thought of as a language,then definitions are the vocabulary and our prior mathematical knowledge indicates ourexperience and versatility with the language.

Mathematics and the English language both share the use of definitions as extremely prac-tical abbreviations. Instead of saying a domesticated carnivorous mammal known scien-tifically as Canis familiaris we would say dog. Instead of writing down there exists aninteger k so that n = km, we write m | n.However, mathematics differs greatly from English in precision and emotional content.Mathematical definitions do not allow ambiguity or sentiment.

Definition 2.4.2

Proposition

A proposition is a true statement that has been proved by a valid argument.

REMARK

You will encounter several variations on the word proposition. A theorem is a particularlysignificant proposition. A lemma is a subsidiary proposition, or more informally, a helperproposition, that is used in the proof of a theorem. A corollary is a proposition that followsalmost immediately from a theorem.

There are particular statements that may look like propositions but are more foundational.An axiom is a statement that is assumed to be true. No proof is given. From axioms wederive propositions and theorems. Obviously, choosing axioms has to be done very carefully.

Consider the following proposition.

Proposition 1 (Transitivity of Divisibility (TD))

Let a, b and c be integers. If a | b and b | c, then a | c.

When one first encounters a proposition, it often helps to work through some examples tounderstand the proof.


Example 7 Suppose a = 3, b = 6 and c = 42. Since 3 | 6 (a | b) and 6 | 42 (b | c), Transitivity ofDivisibility allows us to conclude that 3 | 42 (a | c).Now you might immediately know that 3 | 42. The strength of this proposition is that itworks for any integers a, b, c that satisfy the condition a | b and b | c, not just for theparticular integers of our example.

Now take a minute to read the following proof of Transitivity of Divisibility.

Proof: Since a | b, there exists an integer r so that ra = b. Since b | c, there exists aninteger s so that sb = c. Substituting ra for b in the previous equation, we get (sr)a = c.Since sr is an integer, a | c.

Though this is a simple proof, other proofs can be difficult to read because of the habitsof writing for professional audiences. Many proofs share the following properties which canbe frustrating for students.

1. Proofs are economical. That is, a proof includes what is needed to verify the truth ofa proposition but nothing more.

2. Proofs do not usually identify the hypothesis and the conclusion.

3. Proofs sometimes omit or combine steps.

4. Proofs do not always explicitly justify steps.

5. Proofs do not reflect the process by which the proof was discovered.

The reader of the proof must be conscious of the hypothesis and conclusion, fill in theomitted parts and justify each step.

REMARK

When you are reading a proof of an implication, do the following.

1. Explicitly identify the hypothesis and the conclusion. If the hypothesis contains nostatements write No explicit hypothesis. At the end of the proof, you should beable to identify where each part of the hypothesis has been used.

2. Explicitly identify the core proof technique. When reading a proof, the reader usuallyworks forward from the hypothesis until the conclusion is reached. Specific techniqueswill be covered later in the course.

3. Record any preliminary material needed, usually definitions or propositions that havealready been proved. Judgement is needed here about how much to include.

4. Justify each step with reference to the definitions, previously proved propositions ortechniques used.

5. Add missing steps where necessary and justify these steps with reference to the defi-nitions, previously proved propositions or techniques used.

Section 2.4 Our First Proof 25

Lets analyze the proof of the Transitivity of Divisibility in detail because it will give ussome sense of how to analyze proofs in general. First, observe that If a | b and b | c, thena | c. is an open sentence, and that the domains for the variables a, b and c are specifiedin the first sentence, Let a, b and c be integers.

Professional mathematicians do all of these things implicitly but for the first part of thiscourse, we will do these things explicitly.

We will do a line by line analysis, so to make our work easier, we will write each sentenceon a separate line.

Proof: (For reference, each sentence of the proof is written on a separate line.)

1. Since a | b, there exists an integer r so that ra = b.2. Since b | c, there exists an integer s so that sb = c.3. Substituting ra for b in the previous equation, we get (sr)a = c.

4. Since sr is an integer, a | c.

Lets analyze the proof. What we do now will seem like overkill but it serves two purposes.It gives practice at justifying every line of a proof, and it gives us a structure that wecan use for other proofs. Lastly, recall that the author is proving an implication. Theauthor assumes that the hypothesis is true, and uses the hypothesis to demonstrate thatthe conclusion is true. Here goes.

Analysis of Proof We begin by explicitly identifying the hypothesis and the conclusion.

Hypothesis: a, b and c are integers. a | b and b | c.Conclusion: a | c.Core Proof Technique: Work forwards from the hypothesis.

Preliminary Material: The definition of divides. An integer m divides an integern, and we write m | n, if there exists an integer k so that n = km.

Sentence 1 Since a | b, there exists an integer r so that ra = b.In this sentence, the author of the proof uses the hypothesis a | b and the definitionof divides.

Sentence 2 Since b | c, there exists an integer s so that sb = c.In this sentence, the author uses the hypothesis b | c and the definition of divides.

Sentence 3 Substituting ra for b in the previous equation, we get (sr)a = c.

Here, the author works forward using arithmetic. The actual work is:

sb = c and ra = b implies s(ra) = c which implies (sr)a = c.


Sentence 4 Since sr is an integer, a | c.Lastly, the author uses the definition of divides. In this case, the m, k and n of thedefinition apply to the a, sr and c of the proof. It is important to note that sr is aninteger, otherwise the definition of divides does not apply.

At the end of each proof, you should be able to identify where each part of the hypothesiswas used. It is obvious where a | b and b | c were used. The hypothesis a, b and c areintegers was needed to allow the author to use the definition of divides.

This completes the analysis of our first proof. Between the readings, lectures, quizzes,assignments and tests, you will work your way through roughly one hundred proofs.

Chapter 3

Discovering Proofs

3.1 Objectives


1. Discover a proof using the Direct Proof technique.

2. Write a proof.

3. Read a proof.

The content objectives are:

1. Prove the Divisibility of Integer Combinations.

2. Prove the Bounds By Divisibility.

3. State the Division Algorithm.

3.2 Discovering a Proof

Discovering a proof of a statement is generally hard. There is no recipe for this, but thereare some tips that may be useful, and as we go on through the course, you will learn specifictechniques. Consider the following proposition.

Proposition 1 (Divisibility of Integer Combinations (DIC))

If a, b and c are integers where a | b and a | c, and x and y are any integers, then a | (bx+cy).

As with our first proposition, lets begin with a numeric example.

27

28 Chapter 3 Discovering Proofs

Example 1 Suppose a = 3, b = 6 and c = 27. Then, for any integers x and y, 3 | (6x+ 27y). That is, 3divides any integer combination of 6 and 27. You might say, Thats obvious. Just take acommon factor of 3 from 6x+ 27y. That is

6x+ 27y = 3(2x+ 9y)

That observation is very suggestive of the proof of the Divisibility of Integer Combinations.

The very first thing to do when proving a statement is to explicitly identify the hypothesisand the conclusion. Lets do that for the Divisibility of Integer Combinations.

Hypothesis: a, b, c Z, a | b and a | c. x, y ZConclusion: a | (bx+ cy)

Since we are proving a statement, not using a statement, we assume that the hypothesisis true, and then demonstrate that the conclusion is true. This straightforward approachis called Direct Proof. However, in actually discovering a proof we do not need to workonly forwards from hypothesis. We can work backwards from the conclusion and meetsomewhere in the middle. When writing the proof we must ensure that we begin with thehypothesis and end with the conclusion.

Whether working forwards or backwards, I find it best to proceed by asking questions.When working backwards, I ask

What mathematical fact would allow me to deduce the conclusion?

For example, in the proposition under consideration I would ask

What mathematical fact would allow me to deduce that a | (bx+ cy)?

The answer tells me what to look for or gives me another statement I can work backwardsfrom. In this case the answer would be

If there exists an integer k so that bx+ cy = ak, then a | (bx+ cy).

Note that the answer makes use of the definition of divides. Lets record this statement aspart of a proof in progress.

Proof in Progress

1. To be completed.

2. Since there exists an integer k so that bx+ cy = ka, then a | (bx+ cy).

Now I could ask the question

How can I find such a k?

Section 3.2 Discovering a Proof 29

The answer is not obvious so lets turn to working forwards from the hypothesis. In thiscase my standard two questions are

Have I seen something like this before?What mathematical fact can I deduce from what I already know?

I have seen a | b in an hypothesis before. Twice actually, once in the proof of the Transitivityof Divisibility and once in the prior example. Just as was done in the proof of the Transitivityof Divisibility, I can use a | b and the definition of divisibility to assert that

There exists an integer r such that b = ra.

and Ill add this to the proof in progress.

Proof in Progress

1. Since a | b, there exists an integer r such that b = ra.2. To be completed.


I also know that a | c so I can use the definition of divisibility again to assert that

There exists an integer s such that c = sa.

and I will add this to the proof in progress as well.

Proof in Progress

1. Since a | b, there exists an integer r such that b = ra.2. Since a | c, there exists an integer s such that c = sa.3. To be completed.


Hmmm, what now? Lets look again at the last sentence. There is a bx + cy in the lastsentence and an algebraic expression for b and c in the first two sentences. Substitutinggives

bx+ cy = (ra)x+ (sa)y

and factoring out the a gives

bx+ cy = (ra)x+ (sa)y = a(rx+ sy)

Does this look familiar? We factored in our numeric example and we are factoring here.If we let k = rx+ sy then, because multiplying integers gives integers and adding integersgives integers, k is an integer. Hence, there exists an integer k so that bx+ cy = ak. Thatis, a | (bx+ cy).We are done. Almost. We have discovered a proof but this is rough work. We must nowwrite a formal proof. Just like any other writing, the amount of detail needed in expressingyour thoughts depends upon the audience. A proof of a statement targeted at an audienceof professional specialists in algebra will not look the same as a proof targeted at a highschool audience. When you approach a proof, you should first make a judgement about theaudience. Write for your peers. That is, write your proof so that you could hand it to aclassmate and expect that they would understand the proof.


Proof: Since a | b, there exists an integer r such that b = ra. Since a | c, there exists aninteger s such that c = sa. Let x and y be any integers. Now bx + cy = (ra)x + (sa)y =a(rx + sy). Since rx + sy is an integer, it follows from the definition of divisibility thata | (bx+ cy).

Note that this proof does not reflect the discovery process, and it is a Direct Proof. Itbegins with the hypothesis and ends with the conclusion.

Before we leave this proposition, lets consider the significance of the hypothesis x and yare integers. Suppose, as in our numeric example, a = 3, b = 6 and c = 27. If we choosex = 3/2 and y = 1/4, ax + by = 45/4 which is not even an integer! This simple exampleemphasizes the importance of the hypothesis.

Exercise 1 Prove the following statement. Let a, b, c and d be integers. If a | c and b | d, then ab | cd.

3.3 Reading A Proof

Here is another proposition and proof.

Proposition 2 (Bounds By Divisibility (BBD))

Let a and b be integers. If a | b and b 6= 0 then |a| |b|.

Proof: Since a | b, there exists an integer q so that b = qa. Since b 6= 0, q 6= 0. But ifq 6= 0, |q| 1. Hence, |b| = |qa| = |q||a| |a|.

Lets analyze this proof. First, we will rewrite the proof line by line.

Proof: (For reference purposes, each sentence of the proof is written on a separate line.)

1. Since a | b, there exists an integer q so that b = qa.2. Since b 6= 0, q 6= 0.3. But if q 6= 0, |q| 1.4. Hence, |b| = |qa| = |q||a| |a|.

Now the analysis.

Section 3.4 The Division Algorithm 31

Analysis of Proof As usual, we begin by explicitly identifying the hypothesis and theconclusion.

Hypothesis: a and b are integers. a | b and b 6= 0.Conclusion: |a| |b|.Core Proof Technique: Direct Proof.

Preliminary Material: The definition of divides.

Now we justify every sentence in the proof.

Sentence 1 Since a | b, there exists an integer q so that b = qa.In this sentence, the author of the proof uses the hypothesis a | b and the definitionof divides.

Sentence 2 Since b 6= 0, q 6= 0.If q were zero, then b = qa would imply that b is zero. Since b is not zero, q cannotbe zero.

Sentence 3 But if q 6= 0, |q| 1.Since q is an integer from Sentence 1, and q is not zero from Sentence 2, q 1 orq 1. In either case, |q| 1.

Sentence 4 Hence, |b| = |qa| = |q||a| |a|.Sentence 1 tells us that b = qa. Taking the absolute value of both sides gives |b| = |qa|and using the properties of absolute values we get |qa| = |q||a|. From Sentence 3,|q| 1 so |q||a| |a|.

3.4 The Division Algorithm

As you have known since grade school, not all integers are divided evenly by other integers.There is usually a remainder. We record this as the Division Algorithm.

Proposition 3 (Division Algorithm (DA))

If a and b are integers, and b > 0, then there exist unique integers q and r such that

a = qb+ r where 0 r < b.

We will not prove this statement now. You will see a proof of the uniqueness part lateron and a complete proof is available in the appendix. [Incomplete: Add to appendix.]Lets see some examples before a few remarks.


Example 2 (Division Algorithm)

a = q b+ r20 = 2 7 + 621 = 3 7 + 020 = 3 7 + 1

REMARK

The integer q is called the quotient. The integer r is called the remainder. The integer r is always strictly less than b. The integer r is always positive or zero. Observe that b | a if and only if the remainder is 0. Though the proposition is commonly known as the Division Algorithm, it is not really

an algorithm since it doesnt provide a finite sequence of steps that will construct qand r.

It turns out that the Division Algorithm is remarkably useful. To see how, we must firstdefine the greatest common divisor, which we do soon.

3.5 Practice

1. Prove the following statement. Let a, b, c Z. If ac | bc and c 6= 0, then a | b.2. Prove the following statement. Let x be an integer. If 2 | (x2 1), then 4 | (x2 1).3. Consider the following statement: If a | 30 then a | 60.

(a) The following proof of the statement is incorrect. Describe what is wrong withthe proof.

Proof: Let a be a divisor of 60. Since a can only contain the prime factors 2, 3,and 5, and since all of these integers are factors of 30 as well, a | 30.

(b) Prove the statement.

Section 3.5 Practice 33

4. Consider the following statement:

Suppose a is an integer. If 32 - ((a2 + 3)(a2 + 7)), then a is even.In trying to prove or disprove this statement, each of parts (a), (b) and (c) containsa flaw. Determine the main flaw in each argument.

(a) Suppose a is even. Then a2 is even, so both a2 + 3 and a2 + 7 are odd. Since 32is even, 32 - ((a2 + 3)(a2 + 7)).

(b) Let a = 1. Then 32|((a2+3)(a2+7)), but a is not even. This is a counterexampleto the statement.

(c) Suppose 32 - ((a2+3)(a2+7)). Since 2|32, 2 - ((a2+3)(a2+7)). This means that(a2 + 3)(a2 + 7) must be odd, so both a2 + 3 and a2 + 7 must be odd. Therefore,a2 is even, and hence a is even.

(d) Prove the statement.

Part II

Foundations

34

Chapter 4

Truth Tables

4.1 Objectives


1. Define not, and, or, implies and if and only if using truth tables.

2. Evaluate logical expressions using truth tables.

3. Use truth tables to establish the equivalence of logical expressions and particularlythe equivalence of the contrapositive and the non-equivalence of the converse.

4.2 Truth Tables as Definitions

Throughout this course we work with statements.

Definition 4.2.1

Statement

A statement is a sentence which is either true or false.

Definition 4.2.2

Compound,Component

All of the statements we need to prove will be compound statements, that is, statementscomposed of several individual statements called component statements.

For example, the compound statement

If a | b and b | c, then a | c.

contains three component statements

a | b,b | c, anda | c

Suppose we let X be the statement a | b and Y be the statement b | c and Z be thestatement a | c. Then our original statement

35

36 Chapter 4 Truth Tables

If a | b and b | c, then a | c.

becomes

X and Y imply Z.

If we knew the truth values of X, Y and Z, then we would be able to determine the truthvalue of the compound statement X and Y imply Z. And that is where truth tables comein. Truth tables contain all possible values of the component statements and determine thetruth value of the compound statement.

Truth tables can be used to define the truth value of a statement or evaluate the truthvalue of a statement. For logical operations like not, and, or, implies and if and only if,truth tables are used to define the truth value of the compound statement.

Definition 4.2.3

NOT

The simplest definition is that of NOT A, written A.

A AT F

F T

In prose, if the statement A is true, then the statement NOT A is false. If the statementA is false, then the statement NOT A is true.

Two very important and common logical connectives are AND and OR. Note that these donot always coincide with our use of the words and and or in the English language!

Definition 4.2.4

AND

The definition of A AND B, written A B, is

A B A BT T T

T F F

F T F

F F F

Definition 4.2.5

OR

The definition of A OR B, written A B, is

A B A BT T T

T F T

F T T

F F F

This is an opportune moment to highlight the difference between mathematical languageand the English language. If you are visiting a friend and your friend offers you coffee or

Section 4.3 Truth Tables to Evaluate Logical Expressions 37

tea, you interpret that to mean that you may have coffee or tea but not both. However,the logical A B results in a true statement when A is true, B is true or both are true. Inmathematics, or is inclusive.

Definition 4.2.6

Implies

The definition of A implies B, written A B, often seems strange.

A B A BT T T

T F F

F T T

F F T

The first two rows in the table make sense. The last two make less sense. How can a falsehypothesis result in a true statement? The basic idea is that if one is allowed to assume anhypothesis which is false, any conclusion can be derived.

We will shortly see that implies is closely related to if and only if.

Definition 4.2.7

If and Only If

The definition of A if and only if B, written A B or A iff B, is

A B A BT T T

T F F

F T F

F F T

4.3 Truth Tables to Evaluate Logical Expressions

We can construct truth tables for compound statements by evaluating parts of the compoundstatement separately and then evaluating the larger statement. Consider the following truthtable which shows the truth values of (AB) for all possible combinations of truth values ofthe component statements A and B. (Brackets serve the same purpose in logical expressionsas they do in arithmetic. They specify the order of operation. In logic the order is: brackets,, , , , , with evaluation from left to right.)

Example 1 Construct a truth table for (A B).

A B A B (A B)T T T F

T F T F

F T T F

F F F T


In the first row of the table A and B are true, so using the definition of or, the statementA B is true. Since the negation of a true statement is false, (A B) is false, whichappears in the last column of the first row. Take a minute to convince yourself that each ofthe remaining rows is correct.

Here is another example.

Example 2 Construct a truth table for A (B C).

A B C B C A (B C)T T T T T

T T F T T

T F T T T

T F F F F

F T T T T

F T F T T

F F T T T

F F F F T

Exercise 1

1. If A,B,C are statements, and A and B are true, and C is false, what is the truthvalue of

(a) A (B C)?(b) A (B C)?(c) A (B C)?(d) (A B) C?

2. Construct a truth table for (A B) C.

Definition 4.3.1

Logicallyequivalent

Two compound statements are logically equivalent if they have the same truth values forall combinations of their component statements. We write S1 S2 to mean S1 is logicallyequivalent to S2.

REMARK

Equivalent statements are enormously useful in proofs. Suppose you wish to prove S1 butare having difficulty. If there is a simpler statement S2 and S1 S2, then you can prove S2instead. In proving S2, you will have proved S1 as well.

Section 4.4 Truth Tables to Evaluate Logical Expressions 39

Example 3 Construct a single truth table for (AB) and (A)(B). Are these statements logicallyequivalent?

A B A B (A B) A B (A) (B)T T T F F F F

T F T F F T F

F T T F T F F

F F F T T T T

Since the columns representing (A B) and (A) (B) are identical,we can conclude that (A B) (A) (B).

The preceding example and your assignments demonstrate DeMorgans Laws.

Proposition 1 (De Morgans Laws (DML))

If A and B are statements, then

1. (A B) (A) (B)2. (A B) (A) (B)

REMARK

The next example shows the equivalence of A B and (A B) (B A). This isparticularly important for proofs. Because A B is equivalent to (A B) (B A),to prove a statement of the form A B, we could prove

1. A B and2. B A.

Example 4 Show that A B is logically equivalent to (A B) (B A).

A B A B A B B A (A B) (B A)T T T T T T

T F F F T F

F T F T F F

F F T T T T

Since the columns representing A B and (A B) (B A) are identical,we can conclude that A B (A B) (B A).


4.4 Contrapositive and Converse

Two particular statements, the contrapositive and the converse, which are derived fromA B, occur frequently in mathematics.

Definition 4.4.1

Contrapositive

The statement B A is called the contrapositive of A B.

We can use truth tables to show that A B B A.

A B A B B A B AT T T F F T

T F F T F F

F T T F T T

F F T T T T

Since the columns representing A B and B A are identical, we can conclude thatA B B A.

REMARK

The logical equivalence of a statement and its contrapositive is extremely useful. If provingA B seems difficult, we could try to prove B A instead. It may be easier!

Definition 4.4.2

Converse

The statement B A is called the converse of A B.

We can use truth tables to show that A B 6 B A.

A B A B B AT T T T

T F F T

F T T F

F F T T

Since the columns representing A B and B A are not identical, we can conclude thatA B 6 B A.

REMARK

It is a common mistake for beginning mathematicians to assume that A B and B Aare the same. They are not! Consider the following statement.

If Lassie is a dog, then Lassie has four legs.

This is a true statement (assuming Lassie has had no amputations or birth defects). Theconverse of this statement is

If Lassie has four legs, then Lassie is a dog.

which is clearly false. Many animals other than dogs have four legs.

Section 4.6 More Examples 41

4.5 More Examples

1. Use a truth table to determine whether or not A (B C) is equivalent to(A B) (A C).

A B C B C A (B C) A B A C (A B) (A C)T T T T T T T T

T T F F T T T T

T F T F T T T T

T F F F T T T T

F T T T T T T T

F T F F F T F F

F F T F F F T F

F F F F F F F F

Since the columns associated with the statements A(BC) and (AB)(AC) areidentical, the two statements are equivalent. That is, A (BC) (AB) (AC).

4.6 Practice

1. Use truth tables to show that for statements A, B and C, the Associativity Lawshold. That is

(a) A (B C) (A B) C(b) A (B C) (A B) C

2. Use truth tables to show that for statements A, B and C, the Distributivity Lawshold. That is

(a) A (B C) (A B) (A C)(b) A (B C) (A B) (A C)

3. Give a logical statement that is equivalent to (A B). Provide evidence in theform of a truth table.

4. Construct a truth table for A B C (A B) (A C).

Chapter 5

Introduction to Sets

5.1 Objectives


1. Define and gain experience with set, element, set-builder notation, defining property,subset, superset, equality of sets, empty set, universal set, complement, cardinality,union, intersection and difference.

2. Be able to read and use Venn diagrams.

5.2 Describing a Set

Sets are foundational in mathematics and literally appear everywhere.

Definition 5.2.1

Set, Element

A set is a collection of objects. The objects that make up a set are called its elements (ormembers).

Sets can contain any type of object. Since this is a math course, we frequently use sets ofnumbers. But sets could contain letters, the letters of the alphabet for example, or books,such as those in a library collection.

It is customary to use uppercase letters (A,B,C . . .) to represent sets and lowercase letters(a, b, c, . . .) to represent elements. If a is an element of the set A, we write a A. If a isnot an element of the set A, we write a 6 A.Small sets can be explicitly listed. For example, the set of even numbers less than 10 is

{2, 4, 6, 8}

Our next set requires prime numbers.

Definition 5.2.2

Prime

An integer p > 1 is called a prime if its only positive divisors are 1 and p; otherwise it iscalled composite.

42

Section 5.2 Describing a Set 43

The set of prime numbers less than 10 is

{2, 3, 5, 7}

When explicitly listing sets, we use curly braces, {}, and separate elements with a comma.Many sets are either too large to be listed (the set of all primes less than 10,000) or aredefined by a rule. In these cases, we employ set-builder notation which makes use of adefining property of the set. For example, the set of all real numbers between 1 and 2inclusive could be written as

{x R | 1 x 2}The part of the description following the bar (|) is the defining property of the set. Someauthors use a colon (:) instead of a bar and write

{x R : 1 x 2}As when explicitly listing sets, we use curly braces, {}.Some letters have become associated with specific sets.

N natural numbers, 1, 2, 3, . . .Z integersQ rational numbers, {pq | p, q Z, q 6= 0}Q irrational numbersR real numbersC complex numbers {x+ yi | x, y R}

Computer scientists begin counting at 0 so the notation N used in a computer sciencecontext likely means the set of integers 0, 1, 2, 3, . . .. Be sure to clarify which set is intended.

Example 1 (Set-Builder Notation)

1. The set of all even integers can be described as

{n Z : 2 | n}There is frequently more than one way of describing a set. Another way of describingthe set of even integers is

{2k | k Z}2. The set of all real solutions to x2 + 4x 2 = 0 can be described as

{x R | x2 + 4x 2 = 0}and, in general, the set of all real solutions to f(x) = 0 can be described as

{x R | f(x) = 0}

3. The set of all positive divisors of 30 can be written as

{n N : n | 30}

4. In calculus, we often use intervals of real numbers. The closed interval [a, b] isdefined as the set

{x R | a x b}

44 Chapter 5 Introduction to Sets

Definition 5.2.3

Subset

A set A is called a subset of a set B, and is written A B, if every element of A belongsto B. Symbolically, we write

A B means x A x B

or equivalentlyA B means For all x A, x B

We sometimes say that A is contained in B.

Example 2{1, 2, 3} {1, 2, 3, 4}

Definition 5.2.4

Proper Subset

A set A is called a proper subset of a set B, and written A B, if every element of Abelongs to B and there exists an element in B which does not belong to A.

In the previous example, it is also the case that

Example 3{1, 2, 3} {1, 2, 3, 4}

Definition 5.2.5

Superset

A set A is called a superset of a set B, and written A B, if every element of B belongsto A. A B is equivalent to B A.

Example 4{1, 2, 3, 4} {1, 2, 3}

Definition 5.2.6

Proper Superset

As before, a set A is called a proper superset of a set B, and written A B, if everyelement of B belongs to A and there exists an element in A which does not belong to B.

Example 5{1, 2, 3, 4} {1, 2, 3}

Definition 5.2.7

Set Equality

Saying that two sets A and B are equal, and writing A = B, means that A and B haveexactly the same elements. The usual way of showing A = B is to show mutual inclusion,that is, show A is contained in B and B is contained in A. Symbolically, we write

A = B means A B AND B A

Section 5.2 Describing a Set 45

Definition 5.2.8

Empty Set

There is a special set, called the empty set and denoted by , which contains no elements.The empty set is a subset of every set.

Definition 5.2.9

Universal Set

When we discuss sets, we are often concerned with subsets of some implicit or specified setU , called the universal set. In our work on divisibility and greatest common divisors, wewill be concerned with integers as the universal set, even when we dont explicitly say so.

Definition 5.2.10

Set Complement

Relative to a universal set U , the complement of a subset A of U , written A, is the set ofall elements in U but not in A. Symbolically, we write

A = {x | x U AND x 6 A} = {x | (x U) (x 6 A)}

Definition 5.2.11

Cardinality

Lastly, the cardinality of a set A, written |A|, is the number of elements in the set.

Example 6 For example, if A = {1, 2, 3, 4}, then |A| = 4. Heres a pair of mind-blowing questions.What is the cardinality of N? How much larger is Q than N?

Example 7 Let S = {x R | x2 = 2} and T = {x Q | x2 = 2}.

1. Describe the set S by listing its elements. What is the cardinality of S?

2. Describe the set T by listing its elements. What is the cardinality of T?

3. List all of the subsets of S.

Solution:

1. S = {2,2}. |S| = 2.2. T = . |T | = 0.3. , {2}, {2}, S


Example 8 Let the universal set for this question be U , the set of natural numbers less than twenty.Let T be the set of integers divisible by three and F be the set of integers divisible by five.

1. Describe T by explicitly listing the set and by using set-builder notation in at leasttwo ways.

2. Find a subset of T of cardinality three.

3. Find an element which belongs to both T and F .

4. Find an element which belongs to neither T nor F .

5. Explicitly list the set T .

Solution:

1. Explicitly listing the set gives T = {3, 6, 9, 12, 15, 18}. Two set-builder descriptions ofthe set are T = {n N : 3 | n, n < 20} and T = {3k | k N, 3k < 20}

2. {3, 6, 9}. There are several choices possible.3. 15. Notice that this is an element, not a set.

4. 1. There are several choices possible.

5. {1, 2, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19}

5.3 Set Operations

Definition 5.3.1

Union

The union of two sets A and B, written AB, is the set of all elements belonging to eitherset A or set B. Symbolically we write

A B = {x | x A OR x B} = {x | (x A) (x B)}

Note that when we say set A or set B we mean the mathematical use of or. That is, theelement can belong to A, B or both A and B.

Definition 5.3.2

Intersection

The intersection of two sets A and B, written A B, is the set of all elements belongingto both set A and set B. Symbolically we write

A B = {x | x A AND x B} = {x | (x A) (x B)}

Section 5.3 Set Operations 47

Definition 5.3.3

Difference

The difference of two sets A and B, written A B (or A \ B), is the set of all elementsbelonging to A but not B. Symbolically we write

AB = {x | x A AND x 6 B} = {x | (x A) (x 6 B)}

If U is the universal set and A U then A = U A.

Example 9 Let the universal set for this question be U , the set of natural numbers less than or equalto twelve. Let T be the set of integers divisible by three, F be the set of integers divisibleby five and P the set of primes. Determine each of the following.

1. T F2. T F3. P

4. P (T F )5. T F6. (T F ) P

Solution:

1. T F = {3, 5, 6, 9, 10, 12}2. T F = 3. P = {1, 4, 6, 8, 9, 10, 12}4. P (T F ) = {3, 5}5. T F = {1, 2, 4, 5, 7, 8, 10, 11}6. (T F ) P = {6, 9, 10, 12}


5.3.1 Venn Diagrams

Venn diagrams can serve as useful illustrations of set relationships. In Figure 5.3.1 below,the universal set is U = {a, b, c, d, e, w}, the set A = {a, b, c, d} and the set B = {d, e}.The element d lies in the intersection of sets A and B. Since d is the only such element,A B = {d}. The element w does not lie in either set A or B.

A B

a

b

cd e

w

Figure 5.3.1: Venn Diagram

[Incomplete: Add schematic Venn diagrams for intersection, union, disjoint,subset, superset, complement]

5.4 Comparing Sets

5.4.1 Sets of Solutions

One common use of sets is to describe values which are solutions to an equation, but carein expression is required here. The following two sentences mean different things.

1. Let a, b, c R, a 6= 0 and b2 4ac 0. The solutions to the quadratic equationax2 + bx+ c = 0

are

x =bb2 4ac

2a

2. Let a, b, c R, a 6= 0 and b2 4ac 0. Then

x =bb2 4ac

2a

are solutions to the quadratic equation

ax2 + bx+ c = 0

The first sentence asserts that a complete description of all solutions is given. The secondsentence only asserts that x = (b b2 4ac)/2a are solutions, not that they are thecomplete solution. In the language of sets, if S is the complete solution to ax2 + bx+ c = 0,and T = {(bb2 4ac)/2a}, Sentence 1 asserts that S = T (which implies S T andT S) but Sentence 2 only asserts that T S.

Section 5.5 Comparing Sets 49

This point can be confusing. Statements about solutions are often implicitly divided intotwo sets: the set S of all solutions and a set T of proposed solutions. One must be carefulto determine whether the statement is equivalent to S = T or T S. Phrases like thesolution or complete solution or all solutions indicate S = T . Phrases like a solution or aresolutions indicate T S.Similar confusion arises when showing that sets have more than one representation. Forexample, a circle centred at the origin O is often defined geometrically as the set of pointsequidistant from O. Others define a circle algebraically in the Cartesian plane as the set ofpoints satisfying x2 + y2 = r2. To show that the two definitions describe the same object,one must show that the two sets of points are equal.

5.4.2 An Example

Given a set S and a set T , there are two very frequent tasks one must perform: one mustshow S T or S = T . In fact, the second task is usually just two instances of the firsttask: to show S = T one can show S T and T S.So, the important message here is that mathematicians must become skilled at demonstrat-ing that S T . The plan in all cases is the same: choose a generic element of S and showthat it belongs to T . Symbolically

S T means x S x Tor equivalently

S T means For all x S, x TThe element chosen must be completely generic and could, if forced, be instantiated as anyelement of the set S. Showing that a specific element of S belongs to T is inadequate.

Example 10 Consider the statement:

Integer multiples of pi are roots of f(x) = (x2 1) sinx.

1. Explicitly identify two sets used in this statement.

2. Are the two sets equal?

3. Is the statement true?

Solution:

1. Let S be the set of all roots of f(x) = (x2 1) sinx. (We could write S moresymbolically as S = {x R | f(x) = 0}.) Let T be the set of integer multiples of pi.(We could also write T more symbolically as T = {npi | n Z}).

2. To show that S = T we must show T S and S T . Since sin(npi) = 0 for allintegers n, we know that f(npi) = 0. Now, the defining property of S is that a realnumber x belongs to S if f(x) = 0. Since f(npi) = 0, npi S. This is equivalent to: ifnpi T then npi S, or equivalently, T S. Now consider x = 1. The value x = 1 isa solution to (x2 1) sinx = 0 and so belongs to S, but it is not an integer multipleof pi, so it does not belong to T . That is, S 6 T and so the two sets are not equal.

3. The statement is true. The statement only claims that T S, not S = T .


5.5 Practice

1. Consider the following proposition.

If A and B are sets, then |A B| = |A|+ |B| |A B|.Complete the following table and verify that the proposition holds for each of thefollowing pairs of sets.

(a) A = {n Z : n | 30} and B = {n Z : n | 42}(b) A = {x R | sinx = 0,2pi x 2pi} and

B = {x R | cosx = 0,2pi x 2pi}|A| |B| |A B| |A B| |A|+ |B| |A B|

(a)

(b)

Chapter 6

More on Sets

6.1 Objectives


1. To gain more experience working with sets.

6.2 Showing Two Sets Are Equal

Lets take a look at two proofs of the same statement about sets. The first uses a chain ofif and only if statements, the second uses mutual inclusion.

Proposition 1 Let A, B and C be arbitrary sets.

A (B C) = (A B) (A C)

Proof: This proof uses a chain of if and only if statements to show that both A (B C)and (A B) (A C) have exactly the same elements. Let x A (B C). Then

x A (B C) (x A) (x (B C)) definition of union (x A) ((x B) (x C)) definition of intersection ((x A) (x B)) ((x A) (x C)) Distributive Law of logic (x A B) (x A C) definition of union x ((A B) (A C)) definition of intersection

Proof: This proof uses mutual inclusion. That is, we will show

1. A (B C) (A B) (A C)

51

52 Chapter 6 More on Sets

2. A (B C) (A B) (A C)

Equivalently, we must show

1. If x A (B C), then x (A B) (A C).2. If x (A B) (A C), then x A (B C).

Let x A(BC). By the definition of union, x A or x (BC). If x A, then by thedefinition of union, x A B and x A C, that is x (A B) (A C). If x B C,then by the definition of intersection, x B and x C. But then by the definition of union,x A B and x A C. Hence, by the definition of intersection, x (A B) (A C).In both cases, x (A B) (A C) as required.Let x (A B) (A C). By the definition of intersection, x A B and x A C. Ifx A, then by the definition of union, x A (B C). If x 6 A, then by the definition ofunion and the fact that x A B, x B. Similarly, x C. But then, by the definitionof intersection, x B C. By the definition of union, x A (B C). In both cases,x A (B C).

The first of these two proofs also uses mutual inclusion. Do you see how?

REMARK

Which technique is better for proving the equality of two sets: a chain of if and only ifstatements or mutual inclusion? Though some of the choice is personal style, the choice isprimarily determined by the reversibility of each step in the proof. A chain of if and onlyif statements only works if each step in the chain is reversible. Thats pretty unusual. Mostof the time when you are proving two sets are equal, you will need to use mutual inclusion.

6.3 More Examples

1. (a) Give a specific example to show that the statement U (S T ) = (U S)Tis false.

(b) Prove the following statement. Let S, T, U be sets. Then

U (S T ) = (U S) (U T )

Solution:

(a) Let U = , S = {1} and T = {2}. Then U (S T ) = and (U S) T = {2}.In this case U (S T ) 6= (U S) T

(b) Proof: To show U (S T ) = (U S) (U T ) we must showi. U (S T ) (U S) (U T ), andii. U (S T ) (U S) (U T ).

This is equivalent to showing

Section 6.3 More Examples 53

i. If x U (S T ), then x (U S) (U T ), andii. If x (U S) (U T ), then x U (S T ).

In the first case, let x U (S T ). By the definition of set intersection, x UAND x ST . If x S, then x U S and so x (U S) (U T ). If x T ,then x U T and so x (U S) (U T ). In either case, x (U S) (U T )as needed.

In the second case, let x (U S) (U T ). By the definition of set unionx U S OR x U T . If x U S, then by the definition of set intersectionx U AND x S. But then x U and x S T so x U (S T ). Ifx U T , then by the definition of set intersection x U AND x T . So again,x U and x S T so x U (S T ). In either case, x x U (S T ).

Part III

Proof Techniques

54

Chapter 7

Quantifiers

7.1 Objectives


1. Learn the basic structure of quantifiers.

2. Learn how to use the Object, Construct and Select Methods.

7.2 Quantifiers

Not all mathematical statements are obviously in the form If A, then B. You will en-counter statements of the form there is, there are, there exists, it has or for all, for each, forevery, for any. The first four are all examples of the existential quantifier there is andthe final four are all examples of the universal quantifier for all. The word existence isused to make it clear that we are looking for or looking at a particular mathematical object.The word universal is used to make it clear that we are looking for or looking at a set ofobjects all of which share some desired behaviour.

REMARK

All statements which use quantifiers are similar to one of the following two statements,though some elements of the sentence may be implicit or appear in a different order.

There exists an x in the set S such that P (x) is true.For every x in the set S, P (x) is true.

where P (x) is an open sentence that uses the variable x.

Some mathematicians prefer a more symbolic approach. The symbol stands for theEnglish expression there exists. The symbol stands for the English expression for all.Symbolically, the two quantified sentences above are written as:

x S, P (x)x S, P (x)

55

56 Chapter 7 Quantifiers

REMARK

All statements which use quantifiers share a basic structure.

1. a quantifier which will be either an existential or universal quantifier,

2. a variable which can be any math

rdw_027 (reading, discovering and writing proofs, version 0.2.7)

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