Basic concepts of mathematics

Zakon E.1-931705-00-3

This book helps the student complete the transition from purely manipulative to rigorous mathematics. The clear exposition covers many topics that are assumed by later courses but are often not covered with any depth or organization: basic set theory, induction, quantifiers, functions and relations, equivalence relations, properties of the real numbers (including consequences of the completeness axiom), fields, and basic properties of n-dimensional Euclidean spaces. The many exercises and optional topics (isomorphism of complete ordered fields, construction of the real numbers through Dedekind cuts, introduction to normed linear spaces, etc.) allow the instructor to adapt this book to many environments and levels of students. Extensive hypertextual cross-references and hyperlinked indexes of terms and notation add truly interactive elements to the text.

---

Table of contents :
Basic Concepts of Mathematics……Page 3
Terms and Conditions……Page 4
Preface……Page 5
Contents……Page 6
About the Author……Page 8
1 Introduction. Sets and their Elements……Page 9
2 Operations on Sets……Page 11
Problems in Set Theory……Page 17
3 Logical Quantifiers……Page 20
4 Relations (Correspondences)……Page 22
Problems in the Theory of Relations……Page 27
5 Mappings……Page 30
Problems on Mappings……Page 34
6 Composition of Relations and Mappings……Page 36
Problems on the Composition of Relations……Page 38
7 Equivalence Relations……Page 40
Problems on Equivalence Relations……Page 43
8 Sequences……Page 45
Problems on Sequences……Page 50
9 Some Theorems on Countable Sets……Page 52
Problems on Countable and Uncountable Sets……Page 56
1 Introduction……Page 58
2 Axioms of an Ordered Field……Page 59
3 Arithmetic Operations in a Field……Page 62
4 Inequalities in an Ordered Field. Absolute Values……Page 65
Problems on Arithmetic Operations and Inequalities in a Field……Page 69
5 Natural Numbers. Induction……Page 70
6 Induction (continued)……Page 75
Problems on Natural Numbers and Induction……Page 78
7 Integers and Rationals……Page 81
Problems on Integers and Rationals……Page 83
8 Bounded Sets in an Ordered Field……Page 84
9 The Completeness Axiom. Suprema and Infima……Page 86
Problems on Bounded Sets, Infima, and Suprema……Page 90
10 Some Applications of the Completeness Axiom……Page 92
Problems on Complete and Archimedean Fields……Page 96
11 Roots. Irrational Numbers……Page 97
Problems on Roots and Irrationals……Page 99
12 Powers with Arbitrary Real Exponents……Page 100
Problems on Powers……Page 103
13 Decimal and other Approximations……Page 105
14 Isomorphism of Complete Ordered Fields……Page 110
Problems on Isomorphisms……Page 117
15 Dedekind Cuts. Construction of E1……Page 118
Problems on Dedekind Cuts……Page 126
16 The Infinities. The lim inf and lim sup of a Sequence……Page 128
Problems on Upper and Lower Limits of Sequences in E*……Page 133
1 Euclidean n-space……Page 135
Problems on Vectors……Page 140
2 Inner Products. Absolute Values. Distances……Page 141
Problems on Vectors (continued)……Page 146
3 Angles and Directions……Page 147
4 Lines and Line Segments……Page 151
Problems on Lines, Angles, and Directions……Page 155
5 Hyperplanes. Linear Functionals……Page 158
Problems on Hyperplanes……Page 163
6 Review Problems on Planes and Lines……Page 166
7 Intervals……Page 170
Problems on Intervals……Page 176
8 Complex Numbers……Page 178
Problems on Complex Numbers……Page 182
9 Vector Spaces. The Space Cn. Euclidean Spaces……Page 184
Problems on Linear Spaces……Page 188
10 Normed Linear Spaces……Page 189
Problems on Normed Linear Spaces……Page 192
Notation……Page 195
Index……Page 196