# How to Prove It: A Structured Approach

Cambridge University Press, 16 de gen. 2006 - 384 pàgines
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.

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### Continguts

 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

### Referències a aquest llibre

 Mathematical Logic for Computer SciencePrevisualització limitada - 2001
 Commonsense ReasoningErik T. MuellerPrevisualització limitada - 2010
Tots els resultats de Google Llibres &raquo;

### Sobre l'autor (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.