怎樣證明數(shù)學題

前言

　　Students of mathematics and computer science often have trouble the first time theyre asked to work seriously with mathematical proofs， because they dont know the "rules of the game." What is expected of you if you are asked to prove something？ What distinguishes a correct proof from an incorrect one？ This book is intended to help students learn the answers to these questions by spelling out the underlying principles involved in the construction of proofs.　　Many students get their first exposure to mathematical proofs in a high school course on geometry. Unfortunately， students in high school geometry are usually taught to think of a proof asia numbered list of statements and reasons， a view of proofs that is too restrictive to be very useful. There is a parallel with computer science here that can be instructive. Early programming languages encouraged a similar restrictive view of computer programs as numbered lists of instructions. Now computer scientists have moved away from such languages and teach programming by using languages that encourage an approach called "structured programming." The discussion of proofs in this book is inspired by the belief that many of the considerations that have led computer scientists to embrace the structured approach to programming apply to proof-writing as well. You might say that this book teaches "structured proving."

內(nèi)容概要

本書介紹了數(shù)學證明的基本要點，內(nèi)容通俗而不失嚴謹，可以幫助高中以上程度的學生熟悉數(shù)學語言，邁入數(shù)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(shù)學、計算機科學、哲學、語言學專業(yè)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(shù)學與計算機科學系教授，《美國數(shù)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(shù)理邏輯，在組合、拓撲、分析、數(shù)學方法論、量子力學等多個領(lǐng)域都發(fā)表了大量論文。

書籍目錄

Introduction1　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 Connectives2　Quantificational Logic　2.1  Quantifiers　2.2  Equivalences Involving Quantifiers　2.3  More Operations on Sets3　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 Proofs4　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 Relations5　Functions　5.1  Functions　5.2  One-to-one and Onto　5.3  Inverses of Functions　5.4  Images and Inverse Images: A Research Project6　Mathematical Induction　6.1  Proof by Mathematical Induction　6.2  More Examples　6.3  Recursion　6.4  Strong Induction　6.5  Closures Again7　Infinite Sets　7.1  Equinumerous Sets　7.2  Countable and Uncountable Sets　7.3  The Cantor-Schr6der-Bernstein TheoremAppendix 1: Solutions to Selected ExercisesAppendix 2: Proof DesignerSuggestions for Further ReadingSummary of Proof TechniquesIndex

章節(jié)摘錄

　　1.1. Deductive Reasoning and Logical Connectives　　As we saw in the introduction, proofs play a central role in mathematics, and deductive reasoning is the foundation on which proofs are based. Therefore, we begin our study of mathematical reasoning and proofs by examining how deductive reasoning works.　　Example 1.1.1. Here are three examples of deductive reasoning:　　1. It will either rain or snow tomorrow.　　Its too warm for snow.　　Therefore, it will rain.　　2. If today is Sunday, then I dont have to go to work today.　　Today is Sunday.　　Therefore, I dont have to go to work today.　　3. I will go to work either tomorrow or today.　　Im going to stay home today.　　Therefore, I will go to work tomorrow.　　In each case, we have arrived at a conclusion from the assumption that some other statements, called premises, are true. For example, the premises in argument 3 are the statements "I will go to work either tomorrow or today" and "Im going to stay home today." The conclusion is "I will go to work tomorrow," and it seems to be forced on us somehow by the premises.

媒體關(guān)注與評論

　　“本書行文簡潔，通俗易懂……習題如此豐富，而且難度各異，層次錯落有致……強烈推薦！”　　—— MAA Reviews　　“本書介紹了數(shù)學證明的基本要點，非常有價值?！薄　　?SIAM Review　　“非常好的一本書！全面而清晰的解釋、豐富的例子、附有解答的習題，使它出類拔萃，但凡你要寫證明，就應(yīng)該選擇它， 無論是自學還是課堂學習?！薄　　狟rent Smith， SIGACT News

編輯推薦

　　《怎樣證明數(shù)學題(英文版·第2版)》深受好評，眾多讀者受益于《怎樣證明數(shù)學題(英文版·第2版)》，學會了如何證明數(shù)學題。無論你來自什么背景，是從事計算機科學還是哲學、語言學，只要你對邏輯和證明感興趣，就應(yīng)該仔細研讀這《怎樣證明數(shù)學題(英文版·第2版)》。研究數(shù)學的師生更是不可錯過《怎樣證明數(shù)學題(英文版·第2版)》。面對證明題，你是否一臉茫然、不知所措呢？是不是迫切需要一個人來教你寫證明呢？《怎樣證明數(shù)學題(英文版·第2版)》將帶給你驚喜，教你一步一步地構(gòu)造證明的框架。閱讀《怎樣證明數(shù)學題(英文版·第2版)》不需要太多的知識背景，只需要你具有高中數(shù)學基礎(chǔ)。為了讓你熟悉數(shù)學語言，作者從構(gòu)建證明的基礎(chǔ)——邏輯和集合論的基本概念講起。豐富的示例，大量的習題，足以讓你在它的指導下掌握證明的“游戲規(guī)則”。新版添加了200多個練習題，并且附錄中給出部分練習的答案或提示。其中一些習題可以用計算機軟件Proof Designer來解答，作者還在附錄中介紹了Proof Designer軟件。

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