How to Prove It: A Structured Approach

by Daniel J. Velleman

• ISBN13: 9780521675994
• ISBN10: 0521675995
• Edition: 2nd
• Format: Paperback
• Copyright: 2006-01-16
• Publisher: Cambridge University Press

Summary

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.

Table of Contents

Preface to the Second Edition	ix
Preface	xi
Introduction	1	(7)
1 Sentential Logic	8	(47)
1.1 Deductive Reasoning and Logical Connectives	8	(6)
1.2 Truth Tables	14	(12)
1.3 Variables and Sets	26	(8)
1.4 Operations on Sets	34	(9)
1.5 The Conditional and Biconditional Connectives	43	(12)
2 Quantificational Logic	55	(29)
2.1 Quantifiers	55	(9)
2.2 Equivalences Involving Quantifiers	64	(9)
2.3 More Operations on Sets	73	(11)
3 Proofs	84	(79)
3.1 Proof Strategies	84	(11)
3.2 Proofs Involving Negations and Conditionals	95	(13)
3.3 Proofs Involving Quantifiers	108	(16)
3.4 Proofs Involving Conjunctions and Biconditionals	124	(12)
3.5 Proofs Involving Disjunctions,	136	(10)
3.6 Existence and Uniqueness Proofs	146	(9)
3.7 More Examples of Proofs	155	(8)
4 Relations	163	(63)
4.1 Ordered Pairs and Cartesian Products	163	(8)
4.2 Relations	171	(9)
4.3 More About Relations	180	(9)
4.4 Ordering Relations	189	(13)
4.5 Closures	202	(11)
4.6 Equivalence Relations	213	(13)
5 Functions	226	(34)
5.1 Functions	226	(10)
5.2 One-to-one and Onto	236	(9)
5.3 Inverses of Functions	245	(10)
5.4 Images and Inverse Images: A Research Project	255	(5)
6 Mathematical Induction	260	(46)
6.1 Proof by Mathematical Induction	260	(7)
6.2 More Examples	267	(12)
6.3 Recursion	279	(9)
6.4 Strong Induction	288	(12)
6.5 Closures Again	300	(6)
7 Infinite Sets	306	(23)
7.1 Equinumerous Sets	306	(9)
7.2 Countable and Uncountable Sets	315	(7)
7.3 The Cantor–Schröder–Bernstein Theorem	322	(7)
Appendix 1: Solutions to Selected Exercises	329	(44)
Appendix 2: Proof Designer	373	(2)
Suggestions for Further Reading	375	(1)
Summary of Proof Techniques	376	(5)
Index	381