How to Prove It: A Structured Approach

Description

Proofs play a central role in advanced mathematics and theoretical computer science, and this bestselling text's third edition will help students transition from solving problems to proving theorems by teaching them the techniques needed to read and write proofs, with a new chapter on number theory and over 150 new exercises. 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 5.1 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-Schro¨der-Bernstein Theorem Appendix: Solutions to Selected Exercises Suggestions for Further Reading Summary of Proof Techniques Index