How to Prove It: A Structured ApproachGeared 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 - LibraryThingThis 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 |
| 375 | |
Summary of Proof Techniques | 376 |
| 381 | |
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Common terms and phrases
According already answer apply arbitrary arbitrary element argument assume assumption Base bound called Chapter choose clearly closure conclude conditional consider contain contradiction correct course defined definition disjoint equation equivalence relation equivalent exactly example exercise f is one-to-one fact false Figure final finite formula function give given goal induction Induction step involving justify least logical forms look mathematical means method natural number notation Note object Once ordered pairs partial order particular person plug positive integer possible problem proof Prove quantifier real number reasoning reflexive relation represented rule Scratch similar Similarly smallest element Solution sometimes stand statement strategy subset Suppose f symbols symmetric Theorem transitive true truth set truth table unique universe variable write written
References to this book
Bibliographic information
| Title | How to Prove It: A Structured Approach |
| Author | Daniel J. Velleman |
| Edition | illustrated, reprint, revised |
| Publisher | Cambridge University Press, 2006 |
| ISBN | 0521675995, 9780521675994 |
| Length | 384 pages |
| Subjects | › Mathematics / General Mathematics / History & Philosophy Mathematics / Logic |
| Export Citation | BiBTeX EndNote RefMan |