出版時(shí)間:2009-07 出版社:人民郵電出版社 作者:Daniel J. Velleman 頁數(shù):384
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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(xué)證明的基本要點(diǎn),內(nèi)容通俗而不失嚴(yán)謹(jǐn),可以幫助高中以上程度的學(xué)生熟悉數(shù)學(xué)語言,邁入數(shù)學(xué)殿堂。新版添加了200多個(gè)練習(xí)題,附錄中給出部分練習(xí)的答案或提示。 本書適用于任何對(duì)邏輯和證明感興趣的人,數(shù)學(xué)、計(jì)算機(jī)科學(xué)、哲學(xué)、語言學(xué)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(xué)院數(shù)學(xué)與計(jì)算機(jī)科學(xué)系教授,《美國數(shù)學(xué)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?fù)?、分析、?shù)學(xué)方法論、量子力學(xué)等多個(gè)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(píng)論
“本書行文簡潔,通俗易懂……習(xí)題如此豐富,而且難度各異,層次錯(cuò)落有致……強(qiáng)烈推薦!” —— MAA Reviews “本書介紹了數(shù)學(xué)證明的基本要點(diǎn),非常有價(jià)值。” —— 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(xí)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(xué)還是課堂學(xué)習(xí)。” ——Brent Smith, SIGACT News
編輯推薦
《怎樣證明數(shù)學(xué)題(英文版·第2版)》深受好評(píng),眾多讀者受益于《怎樣證明數(shù)學(xué)題(英文版·第2版)》,學(xué)會(huì)了如何證明數(shù)學(xué)題。無論你來自什么背景,是從事計(jì)算機(jī)科學(xué)還是哲學(xué)、語言學(xué),只要你對(duì)邏輯和證明感興趣,就應(yīng)該仔細(xì)研讀這《怎樣證明數(shù)學(xué)題(英文版·第2版)》。研究數(shù)學(xué)的師生更是不可錯(cuò)過《怎樣證明數(shù)學(xué)題(英文版·第2版)》。面對(duì)證明題,你是否一臉茫然、不知所措呢?是不是迫切需要一個(gè)人來教你寫證明呢?《怎樣證明數(shù)學(xué)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(xué)題(英文版·第2版)》不需要太多的知識(shí)背景,只需要你具有高中數(shù)學(xué)基礎(chǔ)。為了讓你熟悉數(shù)學(xué)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(xí)題,足以讓你在它的指導(dǎo)下掌握證明的“游戲規(guī)則”。新版添加了200多個(gè)練習(xí)題,并且附錄中給出部分練習(xí)的答案或提示。其中一些習(xí)題可以用計(jì)算機(jī)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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