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222 Cardinality of Sets Here is a significant fact: The elements of any countably infinite set A can be written in an infinitely long list a 1 , a 2 , a 3 , a 4 ,... that begins with some element a 1 ∈ A and includes every element of A. For example, the set E in the above example can be written in list form as 0,2,−2,4,−4,6,−6,8,−8,... The reason that this can be done is as follows. Since A is countably infinite, Definition 13.2 says there is a bijection f : N → A. This allows us to list out the set A as an infinite list f (1), f (2), f (3), f (4),... Conversely, if the elements of A can be written in list form as a 1 , a 2 , a 3 ,..., then the function f : N → A defined as f (n) = a n is a bijection, so A is countably infinite. We summarize this as follows. Theorem 13.3 A set A is countably infinite if and only if its elements can be arranged in an infinite list a 1 , a 2 , a 3 , a 4 ,... As an example of how this theorem might be used, let P denote the set of all prime numbers. Since we can list its elements as 2,3,5,7,11,13,..., it follows that the set P is countably infinite. As another consequence of Theorem 13.3, note that we can interpret the fact that the set R is not countably infinite as meaning that it is impossible to write out all the elements of R in an infinite list. (After all, we tried to do that in the table on page 217, and failed!) This raises a question. Is it also impossible to write out all the elements of Q in an infinite list? In other words, is the set Q of rational numbers countably infinite or uncountable? If you start plotting the rational numbers on the number line, they seem to mostly fill up R. Sure, some numbers such as 2, π and e will not be plotted, but the dots representing rational numbers seem to predominate. We might thus expect Q to be uncountable. However, it is a surprising fact that Q is countable. The proof presented below arranges all the rational numbers in an infinitely long list. Theorem 13.4 The set Q of rational numbers is countably infinite. Proof. To prove this, we just need to show how to write the set Q in list form. Begin by arranging all rational numbers in an infinite array. This is done by making the following chart. The top row has a list of all integers, beginning with 0, then alternating signs as they increase. Each column headed by an integer k contains all the fractions (in reduced form) with numerator k. For example, the column headed by 2 contains the fractions 2 1 , 2 3 , 2 5 , 2 7 ,..., and so on. It does not contain 2 2 , 2 4 , 2 6 , etc., because those are not reduced, and in fact their reduced forms appear in the column headed by 1. You should examine this table and convince yourself that it contains all rational numbers in Q.
Countable and Uncountable Sets 223 0 1 −1 2 −2 3 −3 4 −4 5 −5 ··· 0 1 1 1 −1 1 2 1 −2 1 3 1 −3 1 4 1 −4 1 5 1 −5 1 ··· 1 2 −1 2 2 3 −2 3 3 2 −3 2 4 3 −4 3 5 2 −5 2 ··· 1 3 −1 3 2 5 −2 5 3 4 −3 4 4 5 −4 5 5 3 −5 3 ··· 1 4 −1 4 2 7 −2 7 3 5 −3 5 4 7 −4 7 5 4 −5 4 ··· 1 5 −1 5 2 9 −2 9 3 7 −3 7 4 9 −4 9 5 6 −5 6 ··· 1 6 −1 6 2 11 −2 11 3 8 −3 8 4 11 −4 11 5 7 −5 7 ··· 1 7 . −1 7 . 2 13 . −2 13 . 3 10 . −3 10 . 4 13 . −4 13 . 5 8 . −5 8 ··· . . .. Next, draw an infinite path in this array, beginning at 0 1 and snaking back and forth as indicated below. Every rational number is on this path. 0 1 −1 2 −2 3 −3 4 −4 5 −5 ··· 0 1 1 1 −1 1 2 1 −2 1 3 1 −3 1 4 1 −4 1 5 1 −5 1 ··· 1 2 −1 2 2 3 −2 3 3 2 −3 2 4 3 −4 3 5 2 −5 2 ··· 1 3 −1 3 2 5 −2 5 3 4 −3 4 4 5 −4 5 5 3 −5 3 ··· 1 4 −1 4 2 7 −2 7 3 5 −3 5 4 7 −4 7 5 4 −5 4 ··· 1 5 −1 5 2 9 −2 9 3 7 −3 7 4 9 −4 9 5 6 −5 6 ··· 1 6 −1 6 2 11 −2 11 3 8 −3 8 4 11 −4 11 5 7 −5 7 ··· 1 7 −1 7 2 13 −2 13 3 10 −3 10 4 13 −4 13 5 8 −5 8 ··· 1 8 . −1 8 . 2 15 . −2 15 . 3 11 . −3 11 . 4 15 . −4 15 . 5 9 . −5 9 ··· . . ..
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Book of Proof Richard Hammack Virgi
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To my students
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v II How to Prove Conditional State
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Preface In writing this book I have
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ix The book is organized into four
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Part I Fundamentals
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4 Sets Some sets are so significant
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6 Sets These last three examples hi
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8 Sets 1.2 The Cartesian Product Gi
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10 Sets We can also take Cartesian
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12 Sets This idea of “making” a
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14 Sets This is a subset C ⊆ R 2
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16 Sets y y y x x x (a) (b) (c) Fig
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18 Sets B A ∪ B A ∩ B A − B A
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20 Sets Example 1.7 If P is the set
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22 Sets We can also think of A ∩
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24 Sets 1.8 Indexed Sets When a mat
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26 Sets Example 1.11 Here the sets
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28 Sets 1.9 Sets that Are Number Sy
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30 Sets Russell’s paradox arises
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32 Logic In proving theorems, we ap
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34 Logic R(f , g) : The function f
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36 Logic 2.2 And, Or, Not The word
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38 Logic To conclude this section,
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40 Logic You can think of P ⇒ Q a
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42 Logic that it’s impossible tha
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44 Logic P if and only if Q. P is a
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46 Logic Notice that when we plug i
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48 Logic There are two pairs of log
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50 Logic Likewise, a statement such
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52 Logic 2.8 More on Conditional St
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54 Logic Example 2.9 Consider Goldb
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56 Logic Example 2.10 Consider nega
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58 Logic Example 2.13 Negate the fo
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60 Logic 2.12 An Important Note It
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62 Counting Occasionally we may get
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64 Counting To answer this question
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66 Counting we can choose any one o
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68 Counting 3.2 Factorials In worki
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70 Counting We close this section w
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72 Counting Definition 3.2 If n and
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74 Counting Fact 3.3 ( ( ) n n! n I
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76 Counting 3.4 Pascal’s Triangle
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78 Counting coefficients 1 2 1. Sim
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80 Counting We seek the number of 3
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82 Counting 5. How many 7-digit bin
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CHAPTER 4 Direct Proof It is time t
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Definitions 87 4.2 Definitions A pr
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Definitions 89 it necessary to defi
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Direct Proof 91 So the setup for di
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Direct Proof 93 Here is another exa
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Direct Proof 95 This proposition te
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Treating Similar Cases 97 Propositi
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Treating Similar Cases 99 19. Suppo
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Contrapositive Proof 101 Since P
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Congruence of Integers 103 Proposit
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Mathematical Writing 105 Propositio
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Mathematical Writing 107 Since a |
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CHAPTER 6 Proof by Contradiction We
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Proving Statements with Contradicti
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Proving Conditional Statements by C
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Some Words of Advice 115 for intege
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Part III More on Proof
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120 Proving Non-Conditional Stateme
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130 Proofs Involving Sets Generally
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132 Proofs Involving Sets In practi
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134 Proofs Involving Sets Example 8
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136 Proofs Involving Sets Thus, sin
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138 Proofs Involving Sets Though it
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140 Proofs Involving Sets If we add
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142 Proofs Involving Sets so that d
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CHAPTER 9 Disproof E ver since Chap
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146 Disproof deciding whether a sta
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148 Disproof Example 9.2 Conjecture
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150 Disproof 9.3 Disproof by Contra
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CHAPTER 10 Mathematical Induction T
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154 Mathematical Induction This pic
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156 Mathematical Induction To round
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158 Mathematical Induction Proposit
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160 Mathematical Induction Strong i
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164 Mathematical Induction We next
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168 Mathematical Induction 13. For
Page 181: Part IV Relations, Functions and Ca
Page 184 and 185: 174 Relations The set R encodes the
Page 186 and 187: 176 Relations Exercises for Section
Page 188 and 189: 178 Relations Example 11.7 Here A =
Page 190 and 191: 180 Relations Example 11.8 Prove th
Page 192 and 193: 182 Relations 11.2 Equivalence Rela
Page 194 and 195: 184 Relations We close this section
Page 196 and 197: 186 Relations 11.3 Equivalence Clas
Page 198 and 199: 188 Relations Notationally, the uni
Page 200 and 201: 190 Relations In a similar vein, [2
Page 202 and 203: 192 Relations Exercises for Section
Page 204 and 205: CHAPTER 12 Functions You know from
Page 206 and 207: 196 Functions value a to the output
Page 208 and 209: 198 Functions To answer this, first
Page 210 and 211: 200 Functions For more concrete exa
Page 212 and 213: 202 Functions To see that g is surj
Page 214 and 215: 204 Functions Although the underlyi
Page 216 and 217: 206 Functions 12.4 Composition You
Page 218 and 219: 208 Functions Proof. First suppose
Page 220 and 221: 210 Functions A B a 1 a 1 b 2 b 2 c
Page 222 and 223: 212 Functions We can check our work
Page 224 and 225: 214 Functions If you continue your
Page 226 and 227: 216 Cardinality of Sets On the othe
Page 228 and 229: 218 Cardinality of Sets There is a
Page 230 and 231: 220 Cardinality of Sets And saying
Page 234 and 235: 224 Cardinality of Sets Beginning a
Page 236 and 237: 226 Cardinality of Sets Exercises f
Page 238 and 239: 228 Cardinality of Sets f : A → P
Page 240 and 241: 230 Cardinality of Sets 13.4 The Ca
Page 242 and 243: 232 Cardinality of Sets and the whi
Page 244 and 245: 234 Cardinality of Sets Here are so
Page 246 and 247: 236 Cardinality of Sets On the face
Page 248 and 249: Solutions Chapter 1 Exercises Secti
Page 250 and 251: 240 Solutions Sketch the following
Page 252 and 253: 242 Solutions 5. Sketch the sets X
Page 254 and 255: 244 Solutions Section 1.8 1. Suppos
Page 256 and 257: 246 Solutions Section 2.4 Without c
Page 258 and 259: 248 Solutions 7. P ⇒ Q = (P∧
Page 260 and 261: 250 Solutions Section 2.10 Negate t
Page 262 and 263: 252 Solutions (b) How many such cod
Page 264 and 265: 254 Solutions 5. How many 7-digit b
Page 266 and 267: 256 Solutions 15. If n ∈ Z, then
Page 268 and 269: 258 Solutions 7. Proposition Suppos
Page 270 and 271: 260 Solutions 27. If a ≡ 0 (mod 4
Page 272 and 273: 262 Solutions 7. If a, b ∈ Z, the
Page 274 and 275: 264 Solutions This expresses a 3 +
Page 276 and 277: 266 Solutions 19. If n ∈ N, then
Page 278 and 279: 268 Solutions Chapter 8 Exercises 1
Page 280 and 281: 270 Solutions 19. Prove that {9 n :
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272 Solutions 3. If n ∈ Z and n 5
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274 Solutions 25. For all a, b, c
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276 Solutions 5. If n ∈ N, then 2
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278 Solutions (2) Now assume the st
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280 Solutions 23. Use induction to
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282 Solutions Thus, with n + 1 line
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284 Solutions a b a b a b a b a b a
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286 Solutions Therefore 3x − 5z i
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288 Solutions Chapter 12 Exercises
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290 Solutions 13. Consider the func
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292 Solutions Section 12.5 Exercise
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294 Solutions 13. Let f : A → B b
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296 Solutions 11. Partition N into
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298 Solutions 7. Prove or disprove:
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300 Index notation, 197 one-to-one,
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