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本書介紹了數學證明的基本要點，內容通俗而不失嚴謹，可以幫助高中以上程度的學生熟悉數學語言，邁入數學殿堂。新版添加了200多個練習題，附錄中給出部分練習的答案或提示。
本書適用于任何對邏輯和證明感興趣的人，數學、計算機科學、哲學、語言學專業的讀者都可以從中獲益匪淺。
本書適用于任何對邏輯和證明感興趣的人，數學、計算機科學、哲學、語言學專業的讀者都可以從中獲益匪淺。
作者簡介
Daniel J. Velleman 艾姆赫斯特（Amherst）學院數學與計算機科學系教授，《美國數學月刊》主編。另著有 Which Way Did The Bicycle Go和Philosophies of Mathematics。他的研究興趣廣泛，主攻數理邏輯，在組合、拓撲、分析、數學方法論、量子力學等多個領域都發表了大量論文。
目次
Introduction
1 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
2 Quantificational Logic
2.1 Quantifiers
2.2 Equivalences Involving Quantifiers
2.3 More Operations on Sets
3 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
4 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
5 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
6 Mathematical Induction
6.1 Proof by Mathematical Induction
6.2 More Examples
6.3 Recursion
6.4 Strong Induction
6.5 Closures Again
7 Infinite Sets
7.1 Equinumerous Sets
7.2 Countable and Uncountable Sets
7.3 The Cantor-Schr6der-Bernstein Theorem
Appendix 1: Solutions to Selected Exercises
Appendix 2: Proof Designer
Suggestions for Further Reading
Summary of Proof Techniques
Index
1 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
2 Quantificational Logic
2.1 Quantifiers
2.2 Equivalences Involving Quantifiers
2.3 More Operations on Sets
3 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
4 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
5 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
6 Mathematical Induction
6.1 Proof by Mathematical Induction
6.2 More Examples
6.3 Recursion
6.4 Strong Induction
6.5 Closures Again
7 Infinite Sets
7.1 Equinumerous Sets
7.2 Countable and Uncountable Sets
7.3 The Cantor-Schr6der-Bernstein Theorem
Appendix 1: Solutions to Selected Exercises
Appendix 2: Proof Designer
Suggestions for Further Reading
Summary of Proof Techniques
Index
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