1、Undergraduate Texts in Mathematics Editorial Board S. Axler K.A. Ribet For further volumes: http:/ SecondEdition Reading,Writing,andProving 1 C A Closer Look at MathematicsISSN 0172-6056 Springer New York Dordrecht Heidelberg London All rights reserved. This work may not be translated or copied in w
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4、art of Springer Science+Business Media () Editorial Board: S. Axler Mathematics Department USA K.A. Ribet Mathematics Department University of California at Berkeley Berkeley, CA 94720 USA San Francisco State University ribetmath.berkeley.edu axlersfsu.edu UlrichDaepp PamelaGorkin ISBN9781441994783e
5、ISBN9781441994790 LibraryofCongressControlNumber:2011931085 DOI10.1007/9781441994790 San Francisco, CA 94132 Lewisburg,PA17837 DepartmentofMathematics BucknellUniversity Lewisburg,PA17837 DepartmentofMathematics BucknellUniversity 10013, USA), except for brief excerpts in connection with reviews or
6、scholarly analysis. Use in connec- Springer Science+Business Media, LLC2011 pgorkinbucknell.edu USA udaeppbucknell.edu USAFor Hannes and MadeleinePreface You are probably about to teach or take a “rst course in proof techniques,” or maybe you just want to learn more about mathematics. No matter what
7、 the reason, a student who wishes to learn the material in this book likes mathematics and we hope to keep it that way. At this point, students have an intuitive sense of why things are true, but not the exposure to the detailed and critical thinking necessary to survive in the mathematical world. W
8、e have written this book to bridge this gap. In our experience, students beginning this course have little training in rigorous mathematical reasoning; they need guidance. At the end, they are where they should be; on their own. Our aim is to teach the students to read, write, and do mathematics ind
9、ependently, and to do it with clarity, precision, and care. If we can maintain the enthusiasm they have for the subject, or even create some along the way, our book has done what it was intended to do. Reading. This book was written for a course we teach to rst- and second-year college students. The
10、 style is informal. A few problems require calculus, but these are identied as such. Students will also need to participate while reading proofs, prodded by questions (such as, “Why?”). Many detailed examples are provided in each chapter. Since we encourage the students to draw pictures, we include
11、many il- lustrations as well. Exercises, designed to teach certain concepts, are also included. These can be used as a basis for class discussion, or preparation for the class. Stu- dents are expected to solve the exercises before moving on to the problems. Com- plete solutions to all of the exercis
12、es are provided at the end of each chapter. Prob- lems of varying degrees of difculty appear at the end of each chapter. Some prob- lems are simply proofs of theorems that students are asked to read and summarize; others supply details to statements in the text. Though many of the remaining prob- le
13、ms are standard, we hope that students will solve some of the unique problems presented in each chapter. Writing. The bad news is that it is not easy to write a proof well. The good news is that with proper instruction, students quickly learn the basics of writing. We try to write in a way that we h
14、ope is worthy of imitation, but we also provide students viiviii Preface with “tips” on writing, ranging from the (what should be) obvious to the insiders preference (“Dont start a sentence with a symbol.”). Proving. How can someone learn to prove mathematical results? There are many theories on thi
15、s. We believe that learning mathematics is the same as learning to play an instrument or learning to succeed at a particular sport. Someone must provide the background: the tips, information on the basic skills, and the insiders “know-how.” Then the student has to practice. Musicians and athletes pr
16、actice hours a day, and its not surprising that most mathematicians do, too. We will provide students with the background; the exercises and problems are there for practice. The instructor observes, guides, teaches and, if need be, corrects. As with anything else, the more a student practices, the b
17、etter she or he will become at solving problems. Using this book. What should be in a book like this one? Even a quick glance at other texts on this subject will tell you that everyone agrees on certain topics: logic, quantiers, basic set theoretic concepts, mathematical induction, and the denition
18、and properties of functions. The depth of coverage is open to debate, of course. We try to cover logic and quantiers fairly quickly, because we believe that students can only fully appreciate the fundamentals of mathematics when they are applied to interesting problems. What is also apparent is that
19、 after these essential concepts, everyone disagrees on what should be included. Even we prefer to vary our approach depending on our students. We have tried to provide enough material for a exible approach. The Minimal Approach. If you need only the basics, cover Chapters 118. (If you assume the wel
20、l ordering principle, or decide to accept the principle of mathe- matical induction without proof, you can also omit Chapters 12 and 13.) The Usual Approach. This approach includes Chapters 118 and Chapters 21 24. (This is easily doable in a standard semester, if the class meets three hours per week
21、.) The Algebra Approach. For an algebraic slant to the course, cover Chapters 118, omitting Chapter 13 and including Chapters 27 and 28. The Analysis Approach. For a slant toward analysis, cover Chapters 123. (In- clude Chapter 24, if time allows. This is what we usually cover in our course.) Includ
22、e as much material from Chapters 25 and 26 as time allows. Students usu- ally enjoy an introduction to metric spaces. Projects. We have included projects intended to let students demonstrate what they can do when they are on their own. We indicate prerequisites for each project, and have tried to va
23、ry them enough that they can be assigned throughout the semester. The results in these projects come from different areas that we nd particularly interesting. Students can be guided to a project at their level. Since there are open-ended parts in each project, students can take these projects as far
24、 as they want. We usually encourage the students to work on these in groups. Notation. A word about some of our symbols is in order here. In an attempt to make this book user-friendly, we indicate the end of a proof with the well-known symbolu t. The end of an example or exercise is designated by .
25、If a problem is used later in the text, we designate it by Problem # . We also have a fair numberPreface ix of “nonproofs.” These are “proofs” with errors, gaps, or both; the students are asked to nd the aw and to x it. We conclude such “proofs” with the symbol u t ? . Every other symbol will be den
26、ed when we introduce you to it. Denitions are incorporated in the text for ease of reading and the terms dened are given in boldface type. Presenting. We also hope that students will make the transition to thinking of themselves as members of a mathematical community. We encourage the students we ha
27、ve in this class to attend talks, give talks, go to conferences, read mathemat- ical books, watch mathematical movies, read journal articles, and talk with their colleagues about the things in this course that interest them. Our (incomplete, but lengthy) list of references should serve a student wel
28、l as a starting point. Each of the projects works well as the basis of a talk for students, and we have included some background material in each section. We begin the chapter on projects with some tips on speaking about mathematics. Whats new in this edition. We have made many changes to the rst ed
29、ition. First, all exercises now have solutions and every chapter, except for the rst, has at least twenty problems of varying difculty. As a result, the text has now roughly twice as many problems than before. As in the rst edition, denitions are incorporated in the text. In this edition, all deniti
30、ons newly introduced in a chapter appear again in a section with formal statements of the new denitions. We have included a detailed description of denitions by recursion and a recursion theorem. Weve added axioms of set theory to the appendix. We have included new projects: one on the axiom of choi
31、ce and one on complex numbers. We have added some interesting pieces to two projects, Picture Proofs and The Best Number of All (and Some Other Pretty Good Ones). Some chapters have been changed or added. The rst editions Chapter 12, which required more of students than previous chapters, has been b
32、roken into two chapters, now enumerated Chapters 12 and 13. If the instructor wishes, it is possible to simply assume the results in Chapter 13 and omit the chapter. We have also included a new chapter, Chapter 24, on the CantorSchr oderBernstein theorem. We feel that this is the proper culmination
33、to Chapters 2123 and a wonderful way to end the course, but be forewarned that it is not an easy chapter. Thanks to many of you who used the text, we were able to pinpoint areas where we could improve many of our explanations, provide more motivation, or present a different perspective. Our goal was
34、 to nd simpler, more precise explanations, and we hope that we have been successful. One new feature of this text that may interest instructors of the course: We have written solutions to every third problem. These are available on our website (see below). Of course, we have updated our reference li
35、st, made corrections to errors that appeared in the rst version, and, most likely, introduced new errors in the second version. We hope you will send us corrections to errors that you nd in the text, as well as any suggestions you have for improvement. We hope that through reading, writing, proving,
36、 and presenting mathematics, we can produce students who will make good colleagues in every sense of the word.x Preface * Acknowledgments. Writing a book is a long process, and we wish to express our gratitude to those who have helped us along the way. We are, of course, grateful to the students at
37、Bucknell University who suffered through the early versions of the manuscript, as well as those who used later versions. Their comments, suggestions, and detection of errors are most appreciated. We thank Andrew Shaffer for help with the illustrations. We also wish to express our thanks to our colle
38、agues and friends, Gregory Adams, Thomas Cassidy, David Farmer, and Paul McGuire, for helpful conversations. We are particularly grateful to Raymond Mortini for his willingness to carefully read (and criticize) the entire text. The book is surely better for it. We also wish to thank our (former) stu
39、dent editor, Brad Parker. We simply cannot over- state the value of Brads careful reading, insightful comments, and his suggestions for better prose. We thank Universit at Bern, Switzerland for support provided dur- ing our sabbaticals. Finally, we thank Hannes and Madeleine Daepp for putting up wit
40、h innitely many dinner conversations about this text. For the second edition, we wish to thank professors Paul Stanford at The Univer- sity of Texas at Dallas, Matthew Daws at the University of Leeds, Raymond Boute at Ghent University, John M. Lee at the University of Washington, and Buster Thelen f
41、or many thoughtful suggestions. In addition to our colleagues who helped us with the rst edition, we are grateful to John Bourke, Emily Dryden, and Allen Schweins- berg for their helpful comments. We wish to thank Peter McNamara, in particular, for spotting errors and inconsistencies, for suggestion
42、s for other references, and for pointing out sections that were potentially confusing for students. Again, we are grateful to all our colleagues and our students who have helped us to make this a better text. * We thank Hannes Daepp for creating a website to accompany the text. This web- site contai
43、ns complete solutions to all problems numbered 3n, where n is a positive integer. It also contains corrections to both editions of the text. http:/www.facstaff.bucknell.edu/udaepp/readwriteprove/ Lewisburg, PA Ulrich Daepp December 2010 Pamela Gorkin Authors e-mails: udaeppbucknell.edu and pgorkinbu
44、cknell.eduContents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii 1 The How, When, and Why of Mathematics . . . . . . . . . . . . . . . . . . . . . . . . 1 Spotlight: George P olya . . . . . . . . . . . . . . . . .
45、. . . . . . . . . . . . . . . . . . . . . . . . . 8 Tips on Doing Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2 Logically Speaking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3 Introducing the Contraposi
46、tive and Converse . . . . . . . . . . . . . . . . . . . . . 25 4 Set Notation and Quantiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 Tips on Quantication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 5 Proof Techniques . .
47、. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 Tips on Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 6 Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
48、 . . . . . . . . . . . . . . . 59 Spotlight: Paradoxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 7 Operations on Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 8 More on Operations on Sets . . . . .
49、 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 9 The Power Set and the Cartesian Product . . . . . . . . . . . . . . . . . . . . . . . . . 89 Tips on Writing Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 10 Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 T
