The Foundations of Analysis A Straightforward Introduction Book 1 Logic Sets and Numbers 1st Edition by Binmore ISBN 0521299152 978-0521299152

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Product details: 

ISBN 10:  0521299152

ISBN 13:  978-0521299152

Author: K.G. Binmore
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.

Table of contents: 

1 Proofs

1.1 What is a proof?
1.5 Mathematical proof
1.7 Obvious
1.8 The interpretation of a mathematical theory
2 Logic (I)

2.1 Statements
2.4 Equivalence
2.6 Not
2.7 And, or
2.10 Implies
2.12 If and only if
2.13 Proof schema
3 Logic (II)

3.1 Predicates and sets
3.4 Quantifiers
3.6 Manipulations with quantifiers
3.10 More on contradictories
3.13 Examples and counter-examples
4 Set operations

4.1 Subsets
4.4 Complements
4.7 Unions and intersections
4.13 Zermelo–Fraenkel set theory
5 Relations

5.1 Ordered pairs

6 Functions

6.1 Formal definition
6.2 Terminology
6.5 Composition
6.6 Binary operations and groups
6.8 Axiom of choice
7 Real numbers (I)

7.1 Introduction
7.2 Real numbers and length
7.3 Axioms of arithmetic
7.6 Some theorems in arithmetic
7.10 Axioms of order
7.13 Intervals
8 Principle of induction

8.1 Ordered fields
8.2 The natural numbers
8.4 Principle of induction
8.7 Inductive definitions
8.10 Properties of ℕ
8.13 Integers
8.14 Rational numbers
9 Real numbers (II)

9.1 Introduction
9.2 The method of exhaustion
9.3 Bounds
9.4 Continuum axiom
9.7 Supremum and infimum
9.11 Dedekind sections
9.13 Powers
9.16 Infinity
9.19 Denseness of the rationals
9.21 Uniqueness of the real numbers
10 Construction of the number systems

10.1 Models
10.2 Basic assumptions

10.3 Natural numbers
10.4 Peano postulates
10.6 Arithmetic and order
10.10 Measuring lengths
10.11 Positive rational numbers
10.13 Positive real numbers
10.16 Negative numbers and displacements
10.17 Real numbers
10.19 Linear and quadratic equations
10.20 Complex numbers
10.22 Cubic equations
10.23 Polynomials
11 Number theory

11.1 Introduction
11.2 Integers
11.3 Division algorithm
11.5 Factors
11.8 Euclid’s algorithm
11.9 Primes
11.12 Prime factorisation theorem
11.13 Rational numbers
11.16 Ruler and compass constructions
11.20 Radicals
11.21 Transcendental numbers
12 Cardinality

12.1 Counting
12.2 Cardinality
12.4 Countable sets
12.14 Uncountable sets
12.17 Decimal expansions
12.20 Transcendental numbers
12.23 Counting the uncountable
12.24 Ordinal numbers
12.25 Cardinals
12.26 Continuum hypothesis

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