How to Prove It: A Structured Approach

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Cambridge University Press, Jan 16, 2006 - Mathematics - 384 pages
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Geared to preparing students to make the transition from solving problems to proving theorems, this text teaches 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. To help students 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. Previous Edition Hb (1994) 0-521-44116-1 Previous Edition Pb (1994) 0-521-44663-5
 

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User Review  - billlund - LibraryThing

This should be required reading for all math majors and those who want to learn how to write formal proofs. It is well written with lots of examples. Read full review

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Such a wonderful readable introductory to proofs with plenty of exercises

Contents

Introduction
1
Sentential Logic
8
12 Truth Tables
14
13 Variables and Sets
26
14 Operations on Sets
34
15 The Conditional and Biconditional Connectives
43
Quantificational Logic
55
22 Equivalences Involving Quantifiers
64
45 Closures
202
46 Equivalence Relations
213
Functions
226
52 Onetoone and Onto
236
53 Inverses of Functions
245
A Research Project
255
Mathematical Induction
260
62 More Examples
267

23 More Operations on Sets
73
Proofs
84
32 Proofs Involving Negations and Conditionals
95
33 Proofs Involving Quantifiers
108
34 Proofs Involving Conjunctions and Biconditionals
124
35 Proofs Involving Disjunctions
136
36 Existence and Uniqueness Proofs
146
37 More Examples of Proofs
155
Relations
163
42 Relations
171
43 More About Relations
180
44 Ordering Relations
189
63 Recursion
279
64 Strong Induction
288
65 Closures Again
300
Infinite Sets
306
72 Countable and Uncountable Sets
315
73 The CantorSchroderBernstein Theorem
322
Solutions to Selected Exercises
329
Proof Designer
373
Suggestions for Further Reading
375
Summary of Proof Techniques
376
Index
381
Copyright

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About the author (2006)

Daniel J. Velleman received his BA at Dartmouth College in 1976 summa cum laude, the highest distinction in mathematics. He received his PhD from the University of Wisconsin, Madison, in 1980 and was an instructor at the University of Texas, Austin, from 1980 to 1983. His other books include Which Way Did the Bicycle Go? (with Stan Wagon and Joe Konhauser, 1996) and Philosophies of Mathematics (with Alexander George, 2002). Among his awards and distinctions are the Lester R. Ford Award for the paper 'Versatile Coins' (with Istvan Szalkai, 1994), and the Carl B. Allendoerfer Award for the paper 'Permutations and Combination Locks' (with Greg Call, 1996). He has been a member of the editorial board for American Mathematical Monthly since 1997 and was Editor of Dolciani Mathematical Expositions from 1999 to 2004. He published papers in the Journal of Symbolic Logic, Annals of Pure and Applied Logic, Transactions of the American Mathematical Society, Proceedings of the American Mathematical Society, American Mathematical Monthly, the Mathematics Magazine, the Mathematical Intelligencer, the Philosophical Review, and the American Journal of Physics.

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