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How to Prove It : A Structured Approach

by
Edition:
2nd
ISBN13:

9780521675994

ISBN10:
0521675995
Format:
Paperback
Pub. Date:
1/16/2006
Publisher(s):
Cambridge University Press
List Price: $33.99

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This is the 2nd edition with a publication date of 1/16/2006.
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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


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