How to prove it : a structured approach
Daniel J. Velleman (Author)
"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 for the natural numbers, relations, functions, and infinite sets. To give students the opportunity to construct their own proofs, this new edition contains more than 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."--Jacket
Print Book, English, 2006
Second edition View all formats and editions
Cambridge University Press, Cambridge, 2006
Textbooks
xiii, 384 pages : illustrations ; 24 cm
9780521861243, 9780521675994, 0521861241, 0521675995
62084309
Introduction
Sentential logic
1.1 Deductive reasoning and logical connectives
1.2 truth tables
1.3 variables and sets
1.4 operations on sets
1.5 The conditional and biconditional connectives
Quantificational logic
2.1 Quantifiers
2.2 Equivalences involving quantifiers
2.3 More operations on sets
Proofs
3.1 proof strategies
3.2 proofs involving negations and conditionals
3.3 Proofs involving quantifiers
3.4 Proofs involving conjunctions and biconditionals
3.5 Proofs involving disjunctions
3.6 Existence and uniqueness proofs
3.7 More examples of proofs
Relations
4.1 Ordered pairs and cartesian products
4.2 Relations
4.3 More about relations
4.4 Ordering relations
4.5 Closures
4.6 Equivalence relations
Functions
5.1 Functions
5.2 One-to-one and onto
5.3 Inverses of functions
5.4 Images and inverse images: a research project
Mathematical induction
6.1 Proof by mathematical induction
6.2 More examples
6.3 Recursion
6.4 Strong induction
6.5 Closures again
Infinite sets
7.1 Equinumerous sets
7.2 Countable and uncountable sets
7.3 The cantor
Schroder
Bernstein theorem
Appendix 1: Solutions to selected exercises
Appendix 2: Proof designer
Suggestions for further reading
Summary for proof techniques
Index
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