RDW_017 - Reading Discovering and Writing Proofs Version...

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Reading, Discovering and Writing Proofs Version 0.1.7 Steven Furino February 1, 2012
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Contents 1 In the beginning 7 1.1 What Makes a Mathematician a Mathematician? . . . . . . . . . . . . 7 1.2 How The Course Works . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.3 Why do we reason formally? . . . . . . . . . . . . . . . . . . . . . . . . 8 2 The First Time 10 2.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.2 The Language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.3 Implications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.4 Our First Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3 Set It Up 18 3.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.2 Describing a Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.3 Set Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.3.1 Venn Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.4 Comparing Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.5 Working With Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 4 I Swear to Tell The Whole Truth 26 4.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 4.2 Truth Tables as Definitions . . . . . . . . . . . . . . . . . . . . . . . . 26 4.3 Truth Tables to Evaluate Logical Expressions . . . . . . . . . . . . . . 28 5 The Undiscovered Country 30 5.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 5.2 Discovering a Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 5.3 Reading A Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 5.4 The Division Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 33 6 To Be or Not To Be 35 6.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 6.2 Quantifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 6.3 The Object Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 6.4 The Construct Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 6.5 The Select Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 6.6 A Non-Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 7 The Greatest Common Divisor 43 7.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 7.2 Greatest Common Divisor . . . . . . . . . . . . . . . . . . . . . . . . . 43 2
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Section 0.0 CONTENTS 3 7.3 Certificate of Correctess . . . . . . . . . . . . . . . . . . . . . . . . . . 47 7.4 The Extended Euclidean Algorithm (EEA) . . . . . . . . . . . . . . . 49 8 Properties Of GCDs 51 8.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 8.2 Some Useful Propositions . . . . . . . . . . . . . . . . . . . . . . . . . 51 9 Linear Diophantine Equations 57 9.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 9.2 The Select Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 9.3 Linear Diophantine Equations . . . . . . . . . . . . . . . . . . . . . . . 59 10 Nested Quantifiers 67 10.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 10.2 Onto . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 10.2.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 10.2.2 Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 10.2.3 Discovering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 10.3 Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 10.3.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 10.3.2 Reading A Limit Proof . . . . . . . . . . . . . . . . . . . . . . 72 10.3.3 Discovering a Limit Proof . . . . . . . . . . . . . . . . . . . . . 74 11 Practice, Practice, Practice: Quantifiers and Sets 76 11.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 11.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 12 Congruence 79 12.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 12.2 Congruences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 12.2.1 Definition of Congruences . . . . . . . . . . . . . . . . . . . . . 79 12.3 Elementary Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 13 Modular Arithmetic 88 13.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 13.2 Modular Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 13.2.1 [0] Z m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 13.2.2 [1] Z m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 13.2.3 Subtraction in Z m . . . . . . . . . . . . . . . . . . . . . . . . . 90 13.2.4 Division in Z m . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 13.3 Extending Equivalencies . . . . . . . . . . . . . . . . . . . . . . . . . . 92 13.4 Fermat’s Little Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 92 14 Linear Congruences 96 14.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 14.2 The Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 14.3 Extending Equivalencies . . . . . . . . . . . . . . . . . . . . . . . . . . 98 14.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 15 Chinese Remainder Theorem 101 15.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 15.2 An Old Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
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4 Chapter 0 CONTENTS 15.3 Chinese Remainder Theorem . . . . . . . . . . . . . . . . . . . . . . . 102 16 Practice, Practice, Practice: Congruences 105 16.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 16.2 Linear and Polynomial Congruences . . . . . . . . . . . . . . . . . . . 105 16.3 Preparing for RSA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 17 The RSA Scheme 111 17.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 17.2 Why Public Key Cryptography? . . . . . . . . . . . . . . . . . . . . . 111 17.3 Implementing RSA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 17.3.1 Setting up RSA . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 17.3.2 Sending a Message . . . . . . . . . . . . . . . . . . . . . . . . . 112 17.3.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 17.3.3 Receiving a Message . . . . . . . . . . . . . . . . . . . . . . . . 113 17.4 Does M = R ? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 17.5 How Secure Is RSA? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 18 Just Say No 117 18.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 18.2 Negating Statements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 18.3 Negating Statements with Quantifiers . . . . . . . . . . . . . . . . . . 119 18.3.1 Counterexamples . . . . . . . . . . . . . . . . . . . . . . . . . . 120 19 Contradiction 122 19.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 19.2 How To Use Contradiction . . . . . . . . . . . . . . . . . . . . . . . . . 122 19.2.1 When To Use Contradiction . . . . . . . . . . . . . . . . . . . . 123 19.2.2 Reading a Proof by Contradiction . . . . . . . . . . . . . . . . 123 19.2.3 Discovering and Writing a Proof by Contradiction . . . . . . . 124 20 Contrapositive 127 20.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 20.2 The Contrapositive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 20.2.1 When To Use The Contrapositive . . . . . . . . . . . . . . . . . 127 20.3 Reading a Proof That Uses the Contrapositive . . . . . . . . . . . . . 128 20.3.1 Discovering and Writing a Proof Using The Contrapositive . . 129 21 Uniqueness 131 21.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 21.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 21.3 Showing X = Y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 21.4 Finding a Contradiction . . . . . . . . . . . . . . . . . . . . . . . . . . 133 21.5 The Division Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 134 22 Induction 136 22.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 22.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 22.3 Principle of Mathematical Induction . . . . . . . . . . . . . . . . . . . 137 22.3.1 Why Does Induction Work? . . . . . . . . . . . . . . . . . . . .
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