The Foundations of Mathematics

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ISBN: 9780470085011 | 0470085010
Cover: Hardcover
Copyright: 4/7/2008

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Finally there's an easy-to-follow book that will help readers succeed in the art of proving theorems. Sibley not only conveys the spirit of mathematics but also uncovers the skills required to succeed. Key definitions are introduced while readers are encouraged to develop an intuition about these concepts and practice using them in problems. With this approach, they'll gain a strong understanding of the mathematical language as they discover how to apply it in order to find proofs.

Preface	p. ix
Part I
Language, Logic, and Sets	p. 3
Logic and Language	p. 3
Implication	p. 12
Quantifiers and Definitions	p. 22
Introduction to Sets	p. 34
Introduction to Number Theory	p. 43
Additional Set Theory	p. 51
Definitions from Chapter 1	p. 59
Algebraic and Order Properties of Number Systems	p. 60
Proofs	p. 62
Proof Format I: Direct Proofs	p. 63
Proof Format II: Contrapositive and Contradiction	p. 73
Proof Format III: Existence, Uniqueness, Or	p. 81
Proof Format IV: Mathematical Induction	p. 93
The Fundamental Theorem of Arithmetic	p. 107
Further Advice and Practice in Proving	p. 109
Proof Formats	p. 120
Functions	p. 121
Definitions, Notation, and Examples	p. 121
Composition, One-to-One, Onto, and Inverses	p. 133
Images and Pre-Images of Sets	p. 140
Definitions from Chapter 3	p. 146
Relations	p. 147
Relations	p. 147
Equivalence Relations	p. 156
Partitions and Equivalence Relations	p. 164
Partial Orders	p. 167
Definitions from Chapter 4	p. 176
Part II
Infinite Sets	p. 181
The Sizes of Sets	p. 181
Countable Sets	p. 188
Uncountable Sets	p. 196
The Axiom of Choice and Its Equivalents	p. 203
Definitions from Chapter 5	p. 211
Introduction to Discrete Mathematics	p. 213
Graph Theory	p. 213
Trees and Algorithms	p. 222
Counting Principles I	p. 232
Counting Principles II	p. 239
Definitions from Chapter 6	p. 246
Introduction to Abstract Algebra	p. 248
Operations and Properties	p. 248
Groups	p. 254
Groups in Geometry	p. 262
Rings and Fields	p. 264
Lattices	p. 272
Homomorphisms	p. 278
Definitions from Chapter 7	p. 288
Introduction to Analysis	p. 291
Real Numbers, Approximations, and Exact Values	p. 291
Zeno's Paradoxes	p. 301
Limits of Functions	p. 303
Continuous Functions and Counterexamples	p. 312
Counterexamples in Rational Analysis	p. 320
Sequences and Series	p. 321
Discrete Dynamical Systems	p. 331
The Intermediate Value Theorem	p. 346
Definitions for Chapter 8	p. 347
Metamathematics and the Philosophy of Mathematics	p. 351
Metamathematics	p. 351
The Philosophy of Mathematics	p. 361
Definitions for Chapter 9	p. 373
The Greek Alphabet	p. 374
Answers: Selected Answers	p. 375
Index	p. 387
List of Symbols	p. 387
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The Foundations of Mathematics

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Table of Contents

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