数学思维入门：掌握阅读与证明技巧指南

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"《如何阅读和证明：数学思维过程入门》是一本专为本科生和研究生设计的补充教材，旨在系统地教授学生如何理解和进行证明。本书的核心内容围绕六个关键步骤展开，帮助读者掌握证明技巧。这些步骤包括： 1. **抽象**：引导学生理解问题的普遍性和形式，将具体问题转化为更抽象的概念，以便于分析和证明。 2. **公理化系统**：阐述如何基于一套基本的公理和定义构建证明体系，这是数学推理的基础。 3. **概括**：学习如何识别并应用一般原则，从特定案例中提炼出通用的证明方法。 4. **总结证明技术**：提供一个技术指南，列出常用证明技巧，并解释何时在何种情况下使用，如通过关键词判断问题类型。 5. **从A到B的工作流程**：指明如何从已知事实（A）出发，运用逻辑推理或反证法（B）来推导结论。 6. **否定的运用**：通过否定某个假设（NOT B）来寻找矛盾，从而证明原命题的真实性。 7. **构造与猜测**：当解决问题时，鼓励学生猜测可能的对象，并通过证明其性质和结果来确认。 8. **双向证明**：不仅从正面建立论据，也从结果出发逆向推理，增强证明的严谨性。 9. **特殊化策略**：通过将一般原理应用到特定对象上，证明其在特定情况下的有效性。作者Daniel Solow，来自美国Case Western Reserve University的Weatherhead School of Management，这本书旨在培养学生的逻辑思维和证明能力，适合用于课堂教学和自我学习。通过阅读和实践书中的方法，读者将能够提高他们的数学思维水平，更好地理解和完成复杂的证明任务。" 这本书不仅是理论教学的工具，也是培养数学思维习惯和批判性思考的重要资源。无论是初学者还是经验丰富的学生，都能从中受益，提升数学问题解决的能力。同时，它强调了证明过程中的策略选择和逻辑清晰的重要性，对于培养未来科学家、工程师以及各行各业需要逻辑分析能力的人才具有显著价值。

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xiv

Acknowledgments for the Fifth Edition

I thank the following reviewers for their useful comments in guiding the

changes to this ﬁfth edition: Jeﬀ Dodd, Mary Bradley, Kimberly J. Presser,

Ray Rosenstrater, and Richard Delaware. I am grateful to the Naval Post-

graduate Scho ol where, as a visiting researcher, I was able to complete much

of the work on this revision. It was there that I met Lieutenant Comman-

der Jay Foraker, Eng Yau Pee, Andrew Swedberg, Eric Tollefson, Anthony

Tvaryanas, and Roger Vaughn, who were dedicated enough to work through

some of the new exercises that are included i n this edition. I am especially

grateful to Edward Agarwala at Case who prepared a ﬁrst draft of the Solu-

tions Manual for this edition and, in so doing, im proved the quality of some of

the exercises. Thanks also to Ralph Grim aldi from the Rose-Hulman Institute

of Technolo gy who, as an accuracy checker, found numerous minor mistakes

that led to a cleaner version of the text and Solutions Manual. I a lso appre-

ciate the help and support I received from my editors at John Wiley: L aurie

Rosatone and Shannon Corliss.

Acknowledgments for the Sixth Edition

I want to thank John Wiley and Sons for their ﬁnancial support in making the

set of my videotaped lectures that students can use to learn the proof tech-

niques in the ﬁrst 15 chapters of this book. I also acknowl edge the technical

staﬀ of the Weatherhead School of Management at Case Western Reserve Uni-

versity for their professional job in producing these videos. I enjoyed the class

participation of the following students from the Montessori High School in

Cleveland who se voices you hear on the videos: Laurel Booth, Eleanor Dick-

inson, Lucas Elliot, and Hannah Irvine. These young adults have a bright

future. I hope these videos provide extra help for students in learni ng this

material. I am also grateful for the help of Edward Aga rwala who worked

through all of the exercises in Chapters 16, 17, and 18 and for making other

useful suggestions for improving Part II of the book.

D. S.

2 CHAPTER 1: THE TRUTH OF IT ALL

to express the same idea in any language, so there are diﬀerent proofs for

the same mathematical fact. The techniques presented here are designed to

get you started and to guide you through a proof. Consequently, this book

describes not only how the techniq ues work but also when each one is likely to

be used and why. Often you will be able to choose a correct technique based

on the form of the problem under consideration. Therefore, when attempting

to create your own proof, l earn to select a technique consciously before wasting

hours trying to ﬁgure out what to do. The more aware you are of your thought

processes, the better.

The ultimate objective, however, is to use your newly acquired skills and

langua ge to discover a nd communicate previo usl y unknown mathematical

truths. The ﬁrst step in this direction is to reach the level of being abl e to

read proofs and develop your own proo fs of already-known facts. Thi s alone

will g ive you a much deeper a nd richer understanding of the mathematical

universe around you.

Anyone with a good knowledge of high school mathematics can read this

book. Advanced students who have seen proofs before can read the ﬁrst two

chapters, skip to the summary Chapter 15, and subsequently read any of the

appendices to see how all the techniques ﬁt together in a speciﬁc mathematical

subject. Each chapter on a particular technique also contains a brief summa ry

at the end that describes how and when to use the technique. The remainder

of this chapter expla ins the types o f relationships to which proofs are applied.

Additional b ooks on proofs a nd advanced mathematical reasoning are listed

in the bibliography at the end of this book.

1.2 WHAT IS A PROOF?

A proof is a convincing argument expressed in the langua ge of mathematics

that a statement is true. All of the foregoing words are important to your

understanding of w hat a proof is. For exampl e, in mathematics, a statement

is a sentence that is either true or false. Some examples fol low:

1. Two parallel lines in a plane have the same slope.

2. 1 = 0.

3. The real number x 6> 0 (x is not greater than 0).

4. There is an angle t such that cos(t) = t.

Observe that statement (1) is true, (2) is false, and (3) is either true or false,

depending on the value o f a variable.

It is perhaps not as obvious that statement (4) is also true. Furthermore,

a statement that appears to be true can, in fact, be false. This is why it is

necessary to do proofs—you will know that a statement is true onl y when

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