Sets, Functions, and Logic

Contents......Page 6
Preface......Page 8
Students Start Here......Page 10
1.1 It’sNotJust Numbers......Page 12
1.2 Mathematical Notation......Page 15
1.3 Making the Invisible Visible......Page 18
1.4 This Is Where You Come In......Page 20
1.5 The Stuff of Modern Mathematics......Page 22
2.1 The Language of Mathematics: Part 1......Page 24
2.2 Properties of the Language......Page 31
2.3 The Language of Mathematics: Part 2......Page 40
2.4 Properties of Quanti.cation......Page 44
2.5 Proofs in Mathematics......Page 51
2.6 The Integers......Page 61
2.7 Mathematical Truth......Page 65
3.1 Sets......Page 68
3.2 Operations on Sets......Page 72
3.3 Real Intervals......Page 79
3.4 Absolute Values......Page 80
3.5 Inequalities......Page 82
3.6 Arbitrary Unions and Intersections......Page 86
3.7 Cartesian Products......Page 89
3.8 The Historical Development of Set Theory......Page 92
4.1 The Function Concept......Page 98
4.2 Examples of Functions......Page 100
4.3 History of the Modern Function Concept......Page 104
4.4 One-One and Onto Functions......Page 106
4.5 Composition and Inverse Functions......Page 111
4.6 Denumerability......Page 115
4.7 Uncountability......Page 119
5.1 Binary Relations......Page 124
5.2 Properties of Relations......Page 126
5.3 Relations as Sets of Ordered Pairs......Page 129
5.4 Relations as Graphs......Page 132
5.5 Equivalence Relations......Page 133
5.6 Functions as Relations......Page 138
5.7 An Example: The Reals......Page 140
5.8 Upper Bounds. Completeness......Page 142
5.9 Sequences......Page 144
No Answers to the Exercises......Page 148
List of Symbols......Page 150
Index......Page 152