How to prove it : a structured approach

Proofs play a central role in advanced mathematics and theoretical computer science, yet many students struggle the first time they take a course in which proofs play a significant role. This bestselling text's third edition helps students transition from solving problems to proving theorems by teaching them the techniques needed to read and write proofs. Featuring over 150 new exercises and a new chapter on number theory, this new edition introduces students to the world of advanced mathematics through the mastery of proofs. The book begins with the basic concepts of logic and set theory to familiarize students with the language of mathematics and how it is interpreted. These concepts are used as the basis for an analysis of techniques that can be used to build up complex proofs step by step, using detailed 'scratch work' sections to expose the machinery of proofs about numbers, sets, relations, and functions. Assuming no background beyond standard high school mathematics, this book will be useful to anyone interested in logic and proofs: computer scientists, philosophers, linguists, and, of course, mathematicians. -- Provided by publisher Half Title page 2 Title page 3 Copyright page 4 Dedication 6 Contents 7 Preface to the Third Edition 9 Introduction 13 1 Sentential Logic 21 1.1 Deductive Reasoning and Logical Connectives 21 1.2 Truth Tables 28 1.3 Variables and Sets 42 1.4 Operations on Sets 53 1.5 The Conditional and Biconditional Connectives 65 2 Quantificational Logic 80 2.1 Quantifiers 80 2.2 Equivalences Involving Quantifiers 91 2.3 More Operations on Sets 103 3 Proofs 118 3.1 Proof Strategies 118 3.2 Proofs Involving Negations and Conditionals 131 3.3 Proofs Involving Quantifiers 145 3.4 Proofs Involving Conjunctions and Biconditionals 166 3.5 Proofs Involving Disjunctions 179 3.6 Existence and Uniqueness Proofs 192 3.7 More Examples of Proofs 203 4 Relations 215 4.1 Ordered Pairs and Cartesian Products 215 4.2 Relations 225 4.3 More About Relations 236 4.4 Ordering Relations 247 4.5 Equivalence Relations 265 5 Functions 282 5.1 Functions 282 5.2 One-to-One and Onto 294 5.3 Inverses of Functions 306 5.4 Closures 317 5.5 Images and Inverse Images: A Research Project 329 6 Mathematical Induction 335 6.1 Proof by Mathematical Induction 335 6.2 More Examples 343 6.3 Recursion 359 6.4 Strong Induction 371 6.5 Closures Again 387 7 Number Theory 395 7.1 Greatest Common Divisors 395 7.2 Prime Factorization 405 7.3 Modular Arithmetic 415 7.4 Euler’s Theorem 427 7.5 Public-Key Cryptography 436 8 Infinite Sets 452 8.1 Equinumerous Sets 452 8.2 Countable and Uncountable Sets 464 8.3 The Cantor-Schrӧder-Bernstein Theorem 474 Appendix: Solutions to Selected Exercises 483 Suggestions for Further Reading 554 Summary of Proof Techniques 555 Index 558 Many students have trouble the first time they take a mathematics course in which proofs play a significant role. This new edition of Velleman's successful text will prepare students to make the transition from solving problems to proving theorems by teaching them the techniques needed to read and write proofs. The book begins with the basic concepts of logic and set theory, to familiarize students with the language of mathematics and how it is interpreted. These concepts are used as the basis for a step-by-step breakdown of the most important techniques used in constructing proofs. The author shows how complex proofs are built up from these smaller steps, using detailed 'scratch work' sections to expose the machinery of proofs about the natural numbers, relations, functions, and infinite sets. To give students the opportunity to construct their own proofs, this new edition contains over 200 new exercises, selected solutions, and an introduction to Proof Designer software. No background beyond standard high school mathematics is assumed. This book will be useful to anyone interested in logic and proofs: computer scientists, philosophers, linguists, and of course mathematicians.