Reading, Discovering and Writing Proofs Version 1.0 c Faculty of Mathematics, University of Waterloo Contents I Introduction to Proof Methods 9 1 In the beginning 10 1.1 What Makes a Mathematician a Mathematician? . 10 1.2 Why Do We Reason Formally? . 10 1.3 Structure of the Course . 12 2 A First Look At Proofs 14 2.1 Objectives . 14 2.2 The Language . 14 2.3 Propositions, Proofs and Axioms . 15 3 Truth Tables and Logical Operators 18 3.1 Objectives . 18 3.2 Compound Statements . 18 3.3 Truth Tables as Definitions . 19 3.3.1 Negating Statements . 19 3.3.2 Conjunctions and Disjunctions . 20 3.4 More Complicated Statements . 21 3.5 Equivalent Logical Expressions . 22 4 Implications and the Direct Proof 26 4.1 Objectives . 26 4.2 Implications: Hypothesis =) Conclusion . 26 4.3 Rules of Inference . 28 4.4 Proving Implications: The Direct Proof . 30 4.5 Negating an Implication . 31 5 Analysis of a Proof 33 5.1 Objectives . 33 5.2 Divisibility of Integers . 33 5.2.1 Understanding the Definition of Divisibility . 34 5.2.2 Transitivity of Divisibility . 34 5.3 Analyzing the Proof of Transitivity of Divisibility . 35 6 Discovering Proofs 37 6.1 Objectives . 37 6.2 Divisibility of Integer Combinations . 37 6.3 Discovering a Proof of Divisibility of Integer Combinations . 38 6.4 Proof of Bounds by Divisibility . 40 2 Section 0.0 CONTENTS 3 II Foundations: Sets and Quantifiers 43 7 Introduction to Sets 44 7.1 Objectives . 44 7.2 Describing a Set . 44 7.2.1 Set-builder Notation . 46 7.3 Set Operations - Unions, Intersections and Set-Differences . 48 7.4 Cartesian Products of Sets . 50 7.4.1 Cartesian Products of the Form S × S . 50 8 Subsets, Set Equality, Converse and If and Only If 52 8.1 Objectives . 52 8.2 Comparing Sets . 52 8.2.1 Concepts related to Subsets . 53 8.3 Showing Two Sets Are Equal . 55 8.3.1 Converse of an Implication . 56 8.3.2 If and Only If Statements . 56 8.3.3 Set Equality and If and Only If Statements . 59 8.4 Discovering: Sets of Solutions . 60 9 Quantifiers 62 9.1 Objectives . 62 9.2 Quantifiers . 62 9.3 The Universal Quantifier . 64 9.3.1 The Select Method . 65 9.4 The Existential Quantifier . 66 9.4.1 The Construct Method . 67 9.5 Negating Quantifiers . 69 9.6 Assuming a Quantified Statement is True . 70 9.6.1 The Substitution Method . 70 9.6.2 The Object Method . 70 10 Nested Quantifiers 73 10.1 Objectives . 73 10.2 Nested Quantifiers . 73 10.2.1 Negating Nested Quantifiers . 75 10.3 More Examples with Nested Quantifiers . 76 10.4 Functions and Surjections . 77 10.4.1 Graphically . 78 10.4.2 Reading a Proof About Surjection . 79 10.4.3 Discovering a Proof About Surjection . 80 III More Proof Techniques 82 11 Contrapositives and Other Proof Techniques 83 11.1 Objectives . 83 11.2 Proof by Contrapositive . 83 11.3 More Complicated Implications . 86 11.3.1 Method of Elimination . 88 11.4 Summary Examples . 88 4 Chapter 0 CONTENTS 12 Proofs by Contradiction 90 12.1 Objectives . 90 12.2 Proof by Contradiction . 90 12.2.1 When to Use Contradiction . 91 12.2.2 A More Substantial Proof by Contradiction . 92 12.2.3 Discovering and Writing a Proof by Contradiction . 93 13 Uniqueness, Injections and the Division Algorithm 96 13.1 Objectives . 96 13.2 Introduction . 96 13.3 Showing X = Y ................................. 97 13.4 Finding a Contradiction . 98 13.5 One-to-one (Injective) . 99 13.5.1 Discovering a proof about injections . 100 13.5.2 Graphically . 101 13.6 The Division Algorithm . 102 14 Simple Induction 104 14.1 Objectives . 104 14.2 Notation . 104 14.2.1 Summation Notation . 104 14.2.2 Product Notation . 105 14.2.3 Recurrence Relations . 106 14.3 Principle of Mathematical Induction . 107 14.3.1 Why Does Induction Work? . 108 14.3.2 Two Examples of Simple Induction . 108 14.3.3 A Different Starting Point . 110 14.4 An Interesting Example . 112 15 Strong Induction 114 15.1 Objectives . 114 15.2 Strong Induction . 114 15.3 More Examples . 118 16 What's Wrong? 120 16.1 Objectives . 120 16.2 Failure Is More Common Than Success . 120 16.3 Some Questions To Ask . 120 16.4 Assuming What You Need To Prove . 121 16.5 Incorrectly Invoking A Proposition . 121 16.6 Examples With A Universal Quantifier . 122 16.7 Counter-Examples With An Existential Quantifier . 123 16.8 Using the Same Variable For Different Objects . 123 16.9 The Converse Is Not the Contrapositive . 124 16.10 Base Cases in Induction Proofs . 125 16.11 Arithmetic and Unusual Cases . 125 16.12 Not Understanding a Definition . 127 Section 0.0 CONTENTS 5 IV Securing Internet Commerce 128 17 The Greatest Common Divisor 129 17.1 Objectives . 129 17.2 Greatest Common Divisor . 129 17.3 Certificate of Correctess . 133 18 The Extended Euclidean Algorithm 136 18.1 Objectives . 136 18.2 The Extended Euclidean Algorithm (EEA) . 136 19 Properties Of GCDs 141 19.1 Objectives . 141 19.2 Some Useful Propositions . 141 19.3 Using Properties of GCD . 146 20 GCD from Prime Factorization 148 20.1 Objectives . ..