Book of Proof - Amazon S3

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298 Solutions 7. Prove or disprove: If there is a injection f : A → B and a surjection g : A → B, then there is a bijection h : A → B. This is true. Here is an outline of a proof. Define a function g ′ : B → A as follows. For each b ∈ B, choose an element x b ∈ g −1 ({x}). (That is, choose an element x b ∈ A for which g(x b ) = b. Now let g ′ : B → A be the function defined as g ′ (b) = x b . Check that g ′ is injective and apply the the Cantor-Bernstein- Schröeder theorem.
Index C(n, k), 72 absolute value, 6 addition principle, 80 and, 36 axiom of foundation, 30 basis step, 154 biconditional statement, 42 bijection, 216 bijective function, 199 byte, 62 Cantor, Georg, 217 Cantor-Bernstein-Schröeder theorem, 232 cardinality, 4, 215 Cartesian plane, 9 Cartesian power, 10 Cartesian product, 8 closed interval, 6 codomain of a function, 196 Cohen, Paul, 235 complement of a set, 19 composite number, 88 composition of functions, 206 conditional statement, 39 conjecture, 145 constructive proof, 126 continuum hypothesis, 235 contrapositive, 100 converse of a statement, 42, 100 corollary, 86 countable set, 221 counterexample, 147 counting, 61 definition, 85 DeMorgan’s laws, 48, 55 difference of sets, 17 disproof, 144 divides, 88 division algorithm, 28, 89 division algortithm, 89 divisor, 88 domain of a function, 196 Doxiadis, Apostolos, 30 element of a set, 3 empty set, 4 entries of a list, 61 equality of functions, 198 equality of lists, 62 equality of sets, 3 equivalence class, 183 equivalence relation, 182 equivalent statements, 121 Euclid, 112, 138 Euler, Leonhard, 105, 140 existence theorem, 123 existential quantifier, 50 existential statement, 123 factorial, 68 false, 32 Fermat’s last theorem, 34 Fermat, Pierre de, 34 Fibonacci sequence, 165 function, 195 range of, 196 bijective, 199, 215 codomain of, 196 composition of, 206 domain of, 196 equality, 198 injective, 199 inverse, 209
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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
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Part IV Relations, Functions and Ca
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174 Relations The set R encodes the
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176 Relations Exercises for Section
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178 Relations Example 11.7 Here A =
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180 Relations Example 11.8 Prove th
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182 Relations 11.2 Equivalence Rela
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184 Relations We close this section
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186 Relations 11.3 Equivalence Clas
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188 Relations Notationally, the uni
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190 Relations In a similar vein, [2
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192 Relations Exercises for Section
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CHAPTER 12 Functions You know from
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196 Functions value a to the output
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198 Functions To answer this, first
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200 Functions For more concrete exa
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202 Functions To see that g is surj
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204 Functions Although the underlyi
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206 Functions 12.4 Composition You
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208 Functions Proof. First suppose
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210 Functions A B a 1 a 1 b 2 b 2 c
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212 Functions We can check our work
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214 Functions If you continue your
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216 Cardinality of Sets On the othe
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218 Cardinality of Sets There is a
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220 Cardinality of Sets And saying
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222 Cardinality of Sets Here is a s
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224 Cardinality of Sets Beginning a
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226 Cardinality of Sets Exercises f
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228 Cardinality of Sets f : A → P
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230 Cardinality of Sets 13.4 The Ca
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232 Cardinality of Sets and the whi
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234 Cardinality of Sets Here are so
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236 Cardinality of Sets On the face
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Solutions Chapter 1 Exercises Secti
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240 Solutions Sketch the following
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242 Solutions 5. Sketch the sets X
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244 Solutions Section 1.8 1. Suppos
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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 :
Page 282 and 283: 272 Solutions 3. If n ∈ Z and n 5
Page 284 and 285: 274 Solutions 25. For all a, b, c
Page 286 and 287: 276 Solutions 5. If n ∈ N, then 2
Page 288 and 289: 278 Solutions (2) Now assume the st
Page 290 and 291: 280 Solutions 23. Use induction to
Page 292 and 293: 282 Solutions Thus, with n + 1 line
Page 294 and 295: 284 Solutions a b a b a b a b a b a
Page 296 and 297: 286 Solutions Therefore 3x − 5z i
Page 298 and 299: 288 Solutions Chapter 12 Exercises
Page 300 and 301: 290 Solutions 13. Consider the func
Page 302 and 303: 292 Solutions Section 12.5 Exercise
Page 304 and 305: 294 Solutions 13. Let f : A → B b
Page 306 and 307: 296 Solutions 11. Partition N into
Page 310 and 311: 300 Index notation, 197 one-to-one,
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