This book helps the student complete the transition from purely manipulative to rigorous mathematics, with topics that cover basic set theory, fields (with emphasis on the real numbers), a review of the geometry of three dimensions, and properties of linear spaces.
Author(s): Elias Zakon
Year: 2005
Language: English
Pages: 208
Basic Concepts of Mathematics......Page 3
Terms and Conditions......Page 4
Contents......Page 5
Preface......Page 7
About the Author......Page 9
1 Introduction. Sets and their Elements......Page 11
2 Operations on Sets......Page 13
Problems in Set Theory......Page 19
3 Logical Quantifiers......Page 22
4 Relations (Correspondences)......Page 24
Problems in the Theory of Relations......Page 29
5 Mappings......Page 32
Problems on Mappings......Page 36
6 Composition of Relations and Mappings......Page 38
Problems on the Composition of Relations......Page 40
7 Equivalence Relations......Page 42
Problems on Equivalence Relations......Page 45
8 Sequences......Page 47
Problems on Sequences......Page 52
9 Some Theorems on Countable Sets......Page 54
Problems on Countable and Uncountable Sets......Page 58
1 Introduction......Page 61
2 Axioms of an Ordered Field......Page 62
3 Arithmetic Operations in a Field......Page 65
4 Inequalities in an Ordered Field. Absolute Values......Page 68
Problems on Arithmetic Operations and Inequalities in a Field......Page 72
5 Natural Numbers. Induction......Page 73
6 Induction (continued)......Page 78
Problems on Natural Numbers and Induction......Page 81
7 Integers and Rationals......Page 84
Problems on Integers and Rationals......Page 86
8 Bounded Sets in an Ordered Field......Page 87
9 The Completeness Axiom. Suprema and Infima......Page 89
Problems on Bounded Sets, Infima, and Suprema......Page 93
10 Some Applications of the Completeness Axiom......Page 95
Problems on Complete and Archimedean Fields......Page 99
11 Roots. Irrational Numbers......Page 100
Problems on Roots and Irrationals......Page 103
12 Powers with Arbitrary Real Exponents......Page 104
Problems on Powers......Page 106
13 Decimal and other Approximations......Page 108
Problems on Decimal and q-ary Approximations......Page 113
14 Isomorphism of Complete Ordered Fields......Page 114
Problems on Isomorphisms......Page 120
15 Dedekind Cuts. Construction of E1......Page 121
Problems on Dedekind Cuts......Page 129
16 The Infinities. The lim inf and lim sup of a Sequence......Page 131
Problems on Upper and Lower Limits of Sequences in E*......Page 136
1 Euclidean n-space......Page 139
Problems on Vectors......Page 144
2 Inner Products. Absolute Values. Distances......Page 145
Problems on Vectors (continued)......Page 150
3 Angles and Directions......Page 151
4 Lines and Line Segments......Page 155
Problems on Lines, Angles, and Directions......Page 159
5 Hyperplanes. Linear Functionals......Page 162
Problems on Hyperplanes......Page 167
6 Review Problems on Planes and Lines......Page 170
7 Intervals......Page 174
Problems on Intervals......Page 180
8 Complex Numbers......Page 182
Problems on Complex Numbers......Page 186
9 Vector Spaces. The Space Cn. Euclidean Spaces......Page 188
Problems on Linear Spaces......Page 192
10 Normed Linear Spaces......Page 193
Problems on Normed Linear Spaces......Page 196
Notation......Page 199
Index......Page 201
