In elementary introductions to mathematical analysis, the treatment of the logical and algebraic foundations of the subject is necessarily rather skeletal. This book attempts to flesh out the bones of such treatment by providing an informal but systematic account of the foundations of mathematical analysis written at an elementary level. This book is entirely self-contained but, as indicated above, it will be of most use to university or college students who are taking, or who have taken, an introductory course in analysis. Such a course will not automatically cover all the material dealt with in this book and so particular care has been taken to present the material in a manner which makes it suitable for self-study. In a particular, there are a large number of examples and exercises and, where necessary, hints to the solutions are provided. This style of presentation, of course, will also make the book useful for those studying the subject independently of taught course.
Author(s): K.G. Binmore
Edition: 1
Year: 1981
Language: English
Pages: 144
CONTENTS ......Page 5
Introduction ......Page 9
1.1 What is a proof? ......Page 11
1.5 Mathematical proof ......Page 13
1.8 The interpretation of a mathematical theory ......Page 15
2.4 Equivalence ......Page 17
2.7 And, or ......Page 18
2.10 Implies ......Page 19
2.12 If and only if ......Page 21
2.13 Proof schema ......Page 22
3.1 Predicates and sets ......Page 24
3.4 Quantifiers ......Page 26
3.6 Manipulations with quantifiers ......Page 27
3.10 More on contradictories ......Page 28
3.13 Examples and counter-examples ......Page 29
4.1 Subsets ......Page 31
4.4 Complements ......Page 32
4.7 Unions and intersections ......Page 33
4.13 Zermelo-Fraenkel set theory ......Page 35
5.2 Cartesian products ......Page 38
5.3 Relations ......Page 39
5.5 Equivalence relations ......Page 40
5.8 Orderings ......Page 41
6.1 Formal definition ......Page 43
6.2 Terminology ......Page 45
6.5 Composition ......Page 49
6.6 Binary operations and groups ......Page 50
6.8 Axiom of choice ......Page 51
7.2 Real numbers and length ......Page 54
7.3 Axioms of arithmetic ......Page 56
7.6 Some theorems in arithmetic ......Page 59
7.10 Axioms of order ......Page 60
7.13 Intervals ......Page 61
8.2 The natural numbers ......Page 64
8.7 Inductive definitions ......Page 66
8.10 Properties of N ......Page 69
8.14 Rational numbers ......Page 70
9.2 The method of exhaustion ......Page 73
9.4 Continuum axiom ......Page 76
9.7 Supremum and infimum ......Page 77
9.11 Dedekind sections ......Page 80
9.13 Powers ......Page 81
9.16 Infinity ......Page 83
9.19 Denseness of the rationals ......Page 84
9.21- Uniqueness of the real numbers ......Page 85
10.1- Models ......Page 88
10.31 Natural numbers ......Page 89
10.61 Arithmetic and order ......Page 90
10.10 Measuring lengths ......Page 93
10.11 Positive rational numbers ......Page 94
10.13 Positive real numbers ......Page 96
10.16 Negative numbers and displacements ......Page 98
10.17 Real numbers ......Page 99
10.19 Linear and quadratic equations ......Page 101
10.20 Complex numbers ......Page 102
10.22 Cubic equations ......Page 104
10.23 Polynomials ......Page 106
11.1 Introduction ......Page 108
11.3 Division algorithm ......Page 110
11.9 Primes ......Page 111
11.13 Rational numbers ......Page 112
11.16 Ruler and compass constructions ......Page 114
11.20 Radicals ......Page 117
11.21 Transcendental numbers ......Page 118
12.1 Counting ......Page 119
12.2 Cardinality ......Page 120
12.4 Countable sets ......Page 122
12.14 Uncountable sets ......Page 128
12.17 Decimal expansions ......Page 129
12.20 Transcendental numbers ......Page 131
12.23 Counting the uncountable ......Page 132
12.24 Ordinal numbers ......Page 134
12.25 Cardinals ......Page 136
12.26 Continuum hypothesis ......Page 137
Notation ......Page 138
Index......Page 139
