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《计算机问题求解》课程参考书籍教材：Undergraduate Texts in Mathematics——Reading, Writing, and Proving（A Closer Look at Mathematics，Second Edition，S. Axler、K.A. Ribet）

1 The How, When, and Why of Mathematics Spotlight: George Polya Tips on Doing Homework 2 Logically Speaking 3 Introducing the Contrapositive and Converse 4 Set Notation and Quantifiers Tips on Quantification 5 Proof Techniques Tips on Definitions …… 10 Relations Tips on Reading Mathematics 11 Partitions Tips on Putting It All Together 12 Order in the Reals 13 Consequences of the Completeness of R Tips: You Solved ItNow What? 14 Functions, Domain, and Range Spotlight: The Definition of Function 15 Functions, One-to-One, and Onto 16 Inverses 17 Images and Inverse Images Spotlight: Minimum or Infimum? 18 Mathematical Induction 19 Sequences 20 Convergence of Sequences of Real Numbers 21 Equivalent Sets 22 Finite Sets and an Infinite Set 23 Countable and Uncountable Sets 24 The Cantor–Schroder–Bernstein Theorem Spotlight: The Continuum Hypothesis ……

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Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii 1 The How, When, and Why of Mathematics . . . . . . . . . . . . . . . . . . . . . . . . 1 Spotlight: George Polya . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 ´ Tips on Doing Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2 Logically Speaking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3 Introducing the Contrapositive and Converse . . . . . . . . . . . . . . . . . . . . . 25 4 Set Notation and Quantifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 Tips on Quantification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 5 Proof Techniques. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 Tips on Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 6 Sets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 Spotlight: Paradoxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 7 Operations on Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 8 More on Operations on Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 9 The Power Set and the Cartesian Product . . . . . . . . . . . . . . . . . . . . . . . . . 89 Tips on Writing Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 10 Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 Tips on Reading Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 11 Partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 Tips on Putting It All Together . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 12 Order in the Reals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 xi

xii Contents 13 Consequences of the Completeness of R. . . . . . . . . . . . . . . . . . . . . . . . . . . 133 Tips: You Solved It. Now What? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 14 Functions, Domain, and Range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 Spotlight: The Definition of Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 15 Functions, One-to-One, and Onto . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 16 Inverses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 17 Images and Inverse Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 Spotlight: Minimum or Infimum? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 18 Mathematical Induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 19 Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 20 Convergence of Sequences of Real Numbers . . . . . . . . . . . . . . . . . . . . . . . 223 21 Equivalent Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 22 Finite Sets and an Infinite Set. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 23 Countable and Uncountable Sets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 24 The Cantor–Schroder–Bernstein Theorem ¨ . . . . . . . . . . . . . . . . . . . . . . . . 261 Spotlight: The Continuum Hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270 25 Metric Spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 26 Getting to Know Open and Closed Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 27 Modular Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301 28 Fermat’s Little Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315 Spotlight: Public and Secret Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320 29 Projects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325 Tips on Talking about Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325 29.1 Picture Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 29.2 The Best Number of All (and Some Other Pretty Good Ones). . . . . . 330 29.3 Set Constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332 29.4 Rational and Irrational Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334 29.5 Irrationality of e and π . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336 29.6 A Complex Project . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338 29.7 When Does f −1 = 1/ f ? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342 29.8 Pascal’s Triangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343 29.9 The Cantor Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346

Chapter 1 The How,When,and Why of Mathematics What is mathematics?Many people think of mathematics (incorrectly)as addition, subtraction,multiplication,and division of numbers.Those with more mathematical training may think of it as dealing with algorithms.But most professional mathe- maticians think of it as much more than that.While we certainly hope that our students will perform algorithms correctly,what we really want is for them to un- derstand three things:how you do something,why it works,and when it works The problems we present to you in this book concentrate on these three goals.If this is the first time you have been asked to prove theorems,you may find this to be quite a challenge.Not only will you be learning how to solve the problem,you will also be learning how to write up the solution.The necessary definitions and background to understand a problem,as well as a general plan of attack,will always be presented in the text.It's up to you to spend the time reading,trying various ap- proaches,rereading,and reapproaching.You will probably be spending more time on fewer exercises than you ever have before.While you are now beyond the stage of being given steps to follow and practice,there are general rules that can assist you in your transition to doing higher mathematics.Many people have written about this subject before.The classic text on how to approach a problem is a wonderful book called How to Solve It by George Polya,[84]. In his text,Polya gives a list of guidelines for solving mathematical problems.He calls his suggestions"the list."We have included the original in the Appendix on page 366.This list has served as a guide for several generations of mathematicians, and we suggest that you let it guide you as well.Here's a closer look at"the list" with some 21st-century modifications. First."Understanding the problem."Easier said than done,of course.What should you do?Make sure you know what all the words mean.You may need to look something up in this book,or you may need to use another book.Look at the statement to figure out carefully what you are given and what you are supposed to figure out.If a picture will help,draw it.Will you be proving something?What? Will you have to obtain an example?Of what?Check all conditions.Will you have to show that something is false?Once you understand what you have to do,you can move on to the next step. U.Daepp and P.Gorkin,Reading.Writing,and Proving:A Closer Look at Mathematics, Undergraduate Texts in Mathematics,DOI 10.1007/978-1-4419-9479-0_1, Springer Science+Business Media,LLC 2011

Chapter 1 The How, When, and Why of Mathematics What is mathematics? Many people think of mathematics (incorrectly) as addition, subtraction, multiplication, and division of numbers. Those with more mathematical training may think of it as dealing with algorithms. But most professional mathematicians think of it as much more than that. While we certainly hope that our students will perform algorithms correctly, what we really want is for them to understand three things: how you do something, why it works, and when it works. The problems we present to you in this book concentrate on these three goals. If this is the first time you have been asked to prove theorems, you may find this to be quite a challenge. Not only will you be learning how to solve the problem, you will also be learning how to write up the solution. The necessary definitions and background to understand a problem, as well as a general plan of attack, will always be presented in the text. It’s up to you to spend the time reading, trying various approaches, rereading, and reapproaching. You will probably be spending more time on fewer exercises than you ever have before. While you are now beyond the stage of being given steps to follow and practice, there are general rules that can assist you in your transition to doing higher mathematics. Many people have written about this subject before. The classic text on how to approach a problem is a wonderful book called How to Solve It by George Polya, [84]. ´ ´ calls his suggestions “the list.” We have included the original in the Appendix on and we suggest that you let it guide you as well. Here’s a closer look at “the list” First. “Understanding the problem.” Easier said than done, of course. What look something up in this book, or you may need to use another book. Look at the figure out. If a picture will help, draw it. Will you be proving something? What? Will you have to obtain an example? Of what? Check all conditions. Will you have to show that something is false? Once you understand what you have to do, you can move on to the next step. 1 should you do? Make sure you know what all the words mean. You may need to page 366. This list has served as a guide for several generations of mathematicians, statement to figure out carefully what you are given and what you are supposed to with some 21st-century modifications. © Springer Science+Business Media, LLC 2011 U. Daepp and P. Gorkin, Reading, Writing, and Proving: A Closer Look at Mathematics, In his text, Polya gives a list of guidelines for solving mathematical problems. He Undergraduate Texts in Mathematics, DOI 10.1007/978-1-4419-9479-0_1

2 I The How,When,and Why of Mathematics Second."Devising a plan."How will you attack the problem?At this point,you understand what must be done (because you have completed Step 1).Have you seen something like it before?If you haven't looked over class notes,haven't read the text,or haven't done the previous homework assignments,the odds are slim that you have seen anything that will be helpful.Do all that first.Look over the text with the problem in mind,read over your notes with the problem in your head,look at previous exercises and theorems that sound similar.Maybe you can use some of the ideas in the proof of a theorem,or maybe you can use a previous homework problem.Mathematics builds on itself and the problems in the text will also.If you are truly stuck,try to answer a simpler,similar question.Once you decide on a method of approach,try it out. Third."Carrying out the plan."Solve the problem.Look at your solution.Is each sentence true?Sometimes it is difficult to catch an error right after you have "found a solution."Put the problem down and come back to it a few hours later.Is each sentence still true? Fourth."Looking back."Polya suggests checking the result and the argument,or even looking for a different proof.If you are allowed(check with your teacher),one really good way to check a proof is to give it to someone else.You can present it to friends.Even if they don't understand a word you are saying,sometimes saying it out loud in a coherent manner will allow you to recognize an error you can't spot when you are reading.If you are permitted to work together,switch proofs and ask your partner for criticism of your proof. When you are convinced that your argument is correct,it is time to write up a correct and neat solution to the problem. Here is an example of the Polya method at work in mathematics;we will decipher a message.A cipher is a system that is used to hide the meaning of a message by replacing the letters of the alphabet by other letters or symbols. Exercise 1.1.The following message is encoded by a shift of the alphabet;that is, every letter is replaced by another one that has been shifted n places further down the alphabet.Once we reach the end of the alphabet,we start over.For instance,if n were7,we would make the replacements a→h,b→i,..,s→z,t→a,..Now the exercise:What does the message below say? PDEO AJYKZEJC WHCKNEPDI EO YWHHAZ W YWAOWN YELDAN. EP EO RANU AWOU PK XNAWG.NECDP? Let's use the ideas from Polya's list to solve this.If you have solved problems like this before,it might be a better exercise for you to try on your own to see how this fits Polya's method before you read on. 1."Understanding the problem."Each sequence of letters with no blank space be- tween the letters represents one word.Each letter is shifted by the same number of places:namely,n.So n is the unknown in this problem and it is what we need to find.Once we know the value of n,we can decipher the whole message.In addition,once we know the meaning of one letter,we can find the value for n

2 1 The How, When, and Why of Mathematics Second. “Devising a plan.” How will you attack the problem? At this point, you understand what must be done (because you have completed Step 1). Have you seen something like it before? If you haven’t looked over class notes, haven’t read the text, or haven’t done the previous homework assignments, the odds are slim that you have seen anything that will be helpful. Do all that first. Look over the text with the problem in mind, read over your notes with the problem in your head, look at previous exercises and theorems that sound similar. Maybe you can use some of the ideas in the proof of a theorem, or maybe you can use a previous homework problem. Mathematics builds on itself and the problems in the text will also. If you are truly stuck, try to answer a simpler, similar question. Once you decide on a method of approach, try it out. Third. “Carrying out the plan.” Solve the problem. Look at your solution. Is each sentence true? Sometimes it is difficult to catch an error right after you have “found a solution.” Put the problem down and come back to it a few hours later. Is each sentence still true? Fourth. “Looking back.” Polya suggests checking the result and the argument, or ´ even looking for a different proof. If you are allowed (check with your teacher), one really good way to check a proof is to give it to someone else. You can present it to friends. Even if they don’t understand a word you are saying, sometimes saying it out loud in a coherent manner will allow you to recognize an error you can’t spot when you are reading. If you are permitted to work together, switch proofs and ask your partner for criticism of your proof. When you are convinced that your argument is correct, it is time to write up a correct and neat solution to the problem. Here is an example of the Polya method at work in mathematics; we will decipher ´ a message. A cipher is a system that is used to hide the meaning of a message by replacing the letters of the alphabet by other letters or symbols. Exercise 1.1. The following message is encoded by a shift of the alphabet; that is, every letter is replaced by another one that has been shifted n places further down the alphabet. Once we reach the end of the alphabet, we start over. For instance, if n were 7, we would make the replacements a → h, b → i, ..., s → z, t → a, .... Now the exercise: What does the message below say? PDEO AJYKZEJC WHCKNEPDI EO YWHHAZ W YWAOWN YELDAN. EP EO RANU AWOU PK XNAWG, NECDP? Let’s use the ideas from Polya’s list to solve this. If you have solved problems ´ like this before, it might be a better exercise for you to try on your own to see how this fits Polya’s method before you read on. ´ 1. “Understanding the problem.” Each sequence of letters with no blank space between the letters represents one word. Each letter is shifted by the same number of places: namely, n. So n is the unknown in this problem and it is what we need to find. Once we know the value of n, we can decipher the whole message. In addition, once we know the meaning of one letter, we can find the value for n

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