Number Systems And The Foundations Of Analysis

by Elliott Mendelson

• ISBN13: 9780486457925
• ISBN10: 0486457923
• Format: Paperback
• Copyright: 2008-12-18
• Publisher: Dover Publications

Summary

Written by a noted expert on logic and set theory, this study of basic number systems explores natural numbers, integers, rational numbers, real numbers, and complex numbers. Geared toward undergraduate and beginning graduate students, it requires minimal mathematical training. Several helpful appendixes supplement the text. The author is Professor of Mathematics at Queens College in New York.

Table of Contents

Chapter 1. Basic Facts and Notions of Logic and Set Theory

1.1 Logical Connectives

1.2 Conditionals

1.3 Biconditionals

1.4 Quantifiers

1.5 Sets

1.6 Membership. Equality and Inclusion of Sets

1.7 The Empty Set

1.8 Union and Intersection

1.9 Difference and Complement

1.10 Power Set

1.11 Arbitrary Unions and Intersections

1.12 Ordered Pairs

1.13 Cartesian Product

1.14 Relations

1.15 Inverse and Composition of Relations

1.16 Reflexivity, Symmetry, and Transitivity

1.17 Equivalence Relations

1.18 Functions

1.19 Functions from A into (Onto) B

1.20 One-One Functions

1.21 Composition of Functions

1.22 Operations

Chapter 2. The Natural Numbers

2.1 Peano Systems

2.2 The Iteration Theorem

2.3 Application of the Iteration Theorem: Addition

2.4 The Order Relation

2.5 Multiplication

2.6 Exponentiation

2.7 Isomorphism, Categoricity

2.8 A Basic Existence Assumption

Supplementary Exercises

Suggestions for Further Reading

Chapter 3. The Integers

3.1 Definition of the Integers

3.2 Addition and Multiplication of Integers

3.3 Rings and Integral Domains

3.4 Ordered Integral Domains

3.5 Greatest Common Divisor, Primes

3.6 Integers Modulo n

3.7 Characteristic of an Integral Domain

3.8 Natural Numbers and Integers of an Integral Domain

3.9 Subdomains, Isomorphisms, Characterizations of the Integers

Supplementary Exercises

Chapter 4. Rational Numbers and Ordered Fields

4.1 Rational Numbers

4.2 Fields

4.3 Quotient field of an Integral Domain

4.4 Ordered Fields

4.5 Subfields. Rational Numbers of a Field.

Chapter 5. The Real Number System

5.1 Inadequacy of the Rationals

5.2 Archimedean Ordered Fields

5.3 Least Upper Bounds and Greatest Lower Bounds

5.4 The Categoricity of the Theory of Complete Ordered Fields

5.5 Convergent Sequences and Cauchy Sequences

5.6 Cauchy Completion. The Real Number System

5.7 Elementary Topology of the Real Number System

5.8 Continuous Functions

5.9 Infinite Series

Appendix A. Equality

Appendix B. Finite Sums and the Sum Notation

Appendix C. Polynomials

Appendix D. Finite, Infinite, and Denumerable Sets. Cardinal Numbers

Appendix E. Axiomatic Set Theory and the Existence of a Peano System

Appendix F. Construction of the Real Numbers via Dedekind Cuts

Appendix G.Complex Numbers

Bibliography

Index of Special Symbols

Index