The Foundations of Mathematics
by Sibley, Thomas Q.Edition: 1st
Format: Hardcover
Pub. Date: 2008-04-07
Publisher(s): Wiley
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Summary
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.
Author Biography
Thomas Q. Sibley is Professor of Mathematics at St. John's University in Collegeville, Minnesota.
Table of Contents
| 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 |
| Table of Contents provided by Ingram. All Rights Reserved. |
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