Number Systems and the Foundations of Analysis - Brossura

Mendelson, Elliott

Brossura

ISBN 10: 0486457923 ISBN 13: 9780486457925

Casa editrice: Dover Pubns, 2008

Riassunto

This study of basic number systems explores natural numbers, integers, rational numbers, real numbers, and complex numbers. Written by a noted expert on logic and set theory, it assumes no background in abstract mathematical thought. Undergraduates and beginning graduate students will find this treatment an ideal introduction to number systems, particularly in terms of its detailed proofs.
Starting with the basic facts and notions of logic and set theory, the text offers an axiomatic presentation of the simplest structure, the system of natural numbers. It proceeds, by set-theoretic methods, to an examination of integers that covers rings and integral domains, ordered integral domains, and natural numbers and integers of an integral domain. A look at rational numbers and ordered fields follows, along with a survey of the real number system that includes considerations of least upper bounds and greatest lower bounds, convergent and Cauchy sequences, and elementary topology. Numerous exercises and several helpful appendixes supplement the text.

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Contenuti

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

Product Description

Geared toward undergraduate and beginning graduate students, this study explores natural numbers, integers, rational numbers, real numbers, and complex numbers. Numerous exercises and appendixes supplement the text. 1973 edition.

Dettagli dell'edizione

• Editore: Dover Pubns
• Data di pubblicazione: 2008
• Lingua: Inglese
• ISBN 10: 0486457923
• ISBN 13: 9780486457925
• Rilegatura: Copertina flessibile
• Numero di pagine: 358
• Contatto del produttore: non disponibile
• Persona responsabile: Soc. Agr. La Raia Ss; http://www.la-raia.it; ; Strada Monterotondo, 79; Novi Ligure; 15067; Italia