Table of Contents
Chapter 1 Introduction 1
1-1 Our program of study 1
1-2 How numbers developed 1
1-3 The mathematician's view of the development of numbers 4
1-4 A word to the reader 4
1-5 Numbers and numerals 5
Chapter 2 The Natural Numbers 6
2-1 Introduction 6
2-2 Axioms 8
2-3 Using the axioms 19
2-4 Subtraction and division 22
2-5 Arithmetic in other bases 24
2-6 Structure and isomorphism 31
Chapter 3 Sets, Variables, and Statement Forms 35
3-1 Sets 35
3-2 Subsets 36
3-3 Variables and statement forms 42
3-4 Unions, intersections, differences, and products 44
Chapter 4 Mappings and Operations 52
4-1 Mapping of a set into a set 52
4-2 Mappings of a set onto a set 54
4-3 One-to-one mappings 56
4-4 Operations on a set 62
4-5 Mathematical systems 63
Chapter 5 Groups 69
5-1 Definition of a group. Examples 69
5-2 Inherent properties of a group 70
5-3 Permutation groups 73
5-4 Isomorphisms 79
Chapter 6 Relations and Partitions 83
6-1 Relations on a set 83
6-2 Properties of relations 86
6-3 Equivalence relations 88
6-4 Partitions 89
6-5 Order relations 93
Chapter 7 The Integers 95
7-1 The relation fu on ∼ N × N 95
7-2 The operations ⊕ and ⊗ on I 97
7-3 The commutativity and associativity of ⊕ and ⊗ 98
7-4 The number system {I; ⊕, ×} 100
7-5 A new notation for the integers 101
7-6 Subtraction and division 110
7-7 A simplified notation for the integers 112
7-8 Integral domains 113
7-9 Congruences 115
7-10 Conclusion 122
Chapter 8 The Rational Numbers 123
8-1 Constructing the rationals 123
8-2 The operations + and × on the rationals 124
8-3 The commutative and associative laws 126
8-4 Subtraction and division 129
8-5 The cancellation laws 131
8-6 The fractions 132
8-7 Ordering the rationals 135
8-8 Fields 141
Chapter 9 The Real Numbers 142
9-1 Introduction 142
9-2 Repeating decimals 142
9-3 Irrational numbers 149
9-4 Sequences of rationals 155
9-5 The real numbers 166
9-6 The infinite decimals 175
9-7 Countability 184
9-8 Completeness of the reals 190
Index 193
