An Introduction to Proof Through Real Analysis - Daniel J Madden

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Résumé :
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Biographie:
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Sommaire:

List of Figures xiii

Preface xv

Introduction xvii

Part I A First Pass at Defining ? 97

1 Beginnings 3

1.1 A naive approach to the natural numbers 3

1.1.1 Preschool: foundations of the natural numbers 3

1.1.2 Kindergarten: addition and subtraction 5

1.1.3 Grade school: multiplication and division 8

1.1.4 Natural numbers: basic properties and theorems 11

1.2 First steps in proof 12

1.2.1 A direct proof 12

1.2.2 Mathematical induction 14

1.3 Problems 17

2 The Algebra of the Natural Numbers 19

2.1 A more sophisticated look at the basics 19

2.1.1 An algebraic approach 21

2.2 Mathematical induction 22

2.2.1 The theorem of induction 24

2.3 Division 27

2.3.1 The division algorithm 27

2.3.2 Odds and evens 30

2.4 Problems 34

3 Integers 37

3.1 The algebraic properties of ? 37

3.1.1 The algebraic definition of the integers 40

3.1.2 Simple results about integers 42

3.1.3 The relationship between ? and ? 45

3.2 Problems 47

4 Rational Numbers 49

4.1 The algebra 49

4.1.1 Surveying the algebraic properties of ? 49

4.1.2 Defining an ordered field 50

4.1.3 Properties of ordered fields 51

4.2 Fractions versus rational numbers 53

4.2.1 In some ways they are different 53

4.2.2 In some ways they are the same 56

4.3 The rational numbers 58

4.3.1 Operations are well defined 58

4.3.2 ? is an ordered field 63

4.4 The rational numbers are not enough 67

4.4.1 ?2 is irrational 67

4.5 Problems 70

5 Ordered Fields 73

5.1 Other ordered fields 73

5.2 Properties of ordered fields 74

5.2.1 The average theorem 74

5.2.2 Absolute values 75

5.2.3 Picturing number systems 78

5.3 Problems 79

6 The Real Numbers 81

6.1 Completeness 81

6.1.1 Greatest lower bounds 81

6.1.2 So what is complete? 82

6.1.3 An alternate version of completeness 84

6.2 Gaps and caps 86

6.2.1 The Archimedean principle 86

6.2.2 The density theorem 87

6.3 Problems 90

6.4 Appendix 93

Part II Logic, Sets, and Other Basics 97

7 Logic 99

7.1 Propositional logic 99

7.1.1 Logical statements 99

7.1.2 Logical connectives 100

7.1.3 Logical equivalence 104

7.2 Implication 105

7.3 Quantifiers 107

7.3.1 Specification 108

7.3.2 Existence 108

7.3.3 Universal 109

7.3.4 Multiple quantifiers 110

7.4 An application to mathematical definitions 111

7.5 Logic versus English 114

7.6 Problems 116

7.7 Epilogue 118

8 Advice for Constructing Proofs 121

8.1 The structure of a proof 121

8.1.1 Syllogisms 121

8.1.2 The outline of a proof 123

8.2 Methods of proof 127

8.2.1 Direct methods 127

8.2.1.1 Understand how to start 127

8.2.1.2 Parsing logical statements 129

8.2.1.3 Mathematical statements to be proved 131

8.2.1.4 Mathematical statements that are assumed to be true 135

8.2.1.5 What do we know and what do we want? 138

8.2.1.6 Construction of a direct proof 138

8.2.1.7 Compound hypothesis and conclusions 139

8.2.2 Alternate methods of proof 139

8.2.2.1 Contrapositive 139

8.2.2.2 Contradiction 142

8.3 An example of a complicated proof 145

8.4 Problems 149

9 Sets 151

9.1 Defining sets 151

9.2 Starting definitions 153

9.3 Set operations 154

9.3.1 Families of sets 157