How to Prove It: A Structured Approach

Description

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. Half Title page Title page Copyright page Dedication Contents Preface to the Third Edition Introduction 1 Sentential Logic 1.1 Deductive Reasoning and Logical Connectives 1.2 Truth Tables 1.3 Variables and Sets 1.4 Operations on Sets 1.5 The Conditional and Biconditional Connectives 2 Quantificational Logic 2.1 Quantifiers 2.2 Equivalences Involving Quantifiers 2.3 More Operations on Sets 3 Proofs 3.1 Proof Strategies 3.2 Proofs Involving Negations and Conditionals 3.3 Proofs Involving Quantifiers 3.4 Proofs Involving Conjunctions and Biconditionals 3.5 Proofs Involving Disjunctions 3.6 Existence and Uniqueness Proofs 3.7 More Examples of Proofs 4 Relations 4.1 Ordered Pairs and Cartesian Products 4.2 Relations 4.3 More About Relations 4.4 Ordering Relations 4.5 Equivalence Relations 5 Functions 5.1 Functions 5.2 One-to-One and Onto 5.3 Inverses of Functions 5.4 Closures 5.5 Images and Inverse Images: A Research Project 6 Mathematical Induction 6.1 Proof by Mathematical Induction 6.2 More Examples 6.3 Recursion 6.4 Strong Induction 6.5 Closures Again 7 Number Theory 7.1 Greatest Common Divisors 7.2 Prime Factorization 7.3 Modular Arithmetic 7.4 Euler’s Theorem 7.5 Public-Key Cryptography 8 Infinite Sets 8.1 Equinumerous Sets 8.2 Countable and Uncountable Sets 8.3 The Cantor-Schrӧder-Bernstein Theorem Appendix: Solutions to Selected Exercises Suggestions for Further Reading Summary of Proof Techniques Index