Book of Proof - Amazon S3

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122 Proving Non-Conditional Statements and back to P. Similarly, one approach to proving the theorem about the n × n matrix would be to prove the conditional statement (a) ⇒ (b), then (b) ⇒ (c), then (c) ⇒ (d), then (d) ⇒ (e), then (e) ⇒ (f ) and finally (f ) ⇒ (a). This pattern is illustrated below. (a) =⇒ (b) =⇒ (c) ⇑ ⇓ (f ) ⇐= (e) ⇐= (d) Notice that if these six implications have been proved, then it really does follow that the statements (a) through (f) are either all true or all false. If one of them is true, then the circular chain of implications forces them all to be true. On the other hand, if one of them (say (c)) is false, the fact that (b) ⇒ (c) is true forces (b) to be false. This combined with the truth of (a) ⇒ (b) makes (a) false, and so on counterclockwise around the circle. Thus to prove that n statements are equivalent, it suffices to prove n conditional statements showing each statement implies another, in circular pattern. But it is not necessary that the pattern be circular. The following schemes would also do the job: (a) =⇒ (b) ⇐⇒ (c) ⇑ ⇓ (f ) ⇐= (e) ⇐⇒ (d) (a) ⇐⇒ (b) ⇐⇒ (c) ⇕ (f ) ⇐⇒ (e) ⇐⇒ (d) But a circular pattern yields the fewest conditional statements that must be proved. Whatever the pattern, each conditional statement can be proved with either direct, contrapositive or contradiction proof. Though we shall not do any of these proofs in this text, you are sure to encounter them in subsequent courses. 7.3 Existence Proofs; Existence and Uniqueness Proofs Up until this point, we have dealt with proving conditional statements or with statements that can be expressed with two or more conditional statements. Generally, these conditional statements have form P(x) ⇒ Q(x). (Possibly with more than one variable.) We saw in Section 2.8 that this can be interpreted as a universally quantified statement ∀ x, P(x) ⇒ Q(x).
Existence Proofs; Existence and Uniqueness Proofs 123 Thus, conditional statements are universally quantified statements, so in proving a conditional statement—whether we use direct, contrapositive or contradiction proof—we are really proving a universally quantified statement. But how would we prove an existentially quantified statement? What technique would we employ to prove a theorem of the following form? ∃ x, R(x) This statement asserts that there exists some specific object x for which R(x) is true. To prove ∃ x, R(x) is true, all we would have to do is find and display an example of a specific x that makes R(x) true. Though most theorems and propositions are conditional (or if-andonly-if) statements, a few have the form ∃ x, R(x). Such statements are called existence statements, and theorems that have this form are called existence theorems. To prove an existence theorem, all you have to do is provide a particular example that shows it is true. This is often quite simple. (But not always!) Here are some examples: Proposition There exists an even prime number. Proof. Observe that 2 is an even prime number. ■ Admittedly, this last proposition was a bit of an oversimplification. The next one is slightly more challenging. Proposition There exists an integer that can be expressed as the sum of two perfect cubes in two different ways. Proof. Consider the number 1729. Note that 1 3 + 12 3 = 1729 and 9 3 + 10 3 = 1729. Thus the number 1729 can be expressed as the sum of two perfect cubes in two different ways. ■ Sometimes in the proof of an existence statement, a little verification is needed to show that the example really does work. For example, the above proof would be incomplete if we just asserted that 1729 can be written as a sum of two cubes in two ways without showing how this is possible. WARNING: Although an example suffices to prove an existence statement, a single example does not prove a conditional statement.
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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
Page 82 and 83: 72 Counting Definition 3.2 If n and
Page 84 and 85: 74 Counting Fact 3.3 ( ( ) n n! n I
Page 86 and 87: 76 Counting 3.4 Pascal’s Triangle
Page 88 and 89: 78 Counting coefficients 1 2 1. Sim
Page 90 and 91: 80 Counting We seek the number of 3
Page 92 and 93: 82 Counting 5. How many 7-digit bin
Page 95 and 96: CHAPTER 4 Direct Proof It is time t
Page 97 and 98: Definitions 87 4.2 Definitions A pr
Page 99 and 100: Definitions 89 it necessary to defi
Page 101 and 102: Direct Proof 91 So the setup for di
Page 103 and 104: Direct Proof 93 Here is another exa
Page 105 and 106: Direct Proof 95 This proposition te
Page 107 and 108: Treating Similar Cases 97 Propositi
Page 109 and 110: Treating Similar Cases 99 19. Suppo
Page 111 and 112: Contrapositive Proof 101 Since P
Page 113 and 114: Congruence of Integers 103 Proposit
Page 115 and 116: Mathematical Writing 105 Propositio
Page 117 and 118: Mathematical Writing 107 Since a |
Page 119 and 120: CHAPTER 6 Proof by Contradiction We
Page 121 and 122: Proving Statements with Contradicti
Page 123 and 124: Proving Conditional Statements by C
Page 125 and 126: Some Words of Advice 115 for intege
Page 127: Part III More on Proof
Page 130 and 131: 120 Proving Non-Conditional Stateme
Page 140 and 141: 130 Proofs Involving Sets Generally
Page 142 and 143: 132 Proofs Involving Sets In practi
Page 144 and 145: 134 Proofs Involving Sets Example 8
Page 146 and 147: 136 Proofs Involving Sets Thus, sin
Page 148 and 149: 138 Proofs Involving Sets Though it
Page 150 and 151: 140 Proofs Involving Sets If we add
Page 152 and 153: 142 Proofs Involving Sets so that d
Page 154 and 155: CHAPTER 9 Disproof E ver since Chap
Page 156 and 157: 146 Disproof deciding whether a sta
Page 158 and 159: 148 Disproof Example 9.2 Conjecture
Page 160 and 161: 150 Disproof 9.3 Disproof by Contra
Page 162 and 163: CHAPTER 10 Mathematical Induction T
Page 164 and 165: 154 Mathematical Induction This pic
Page 166 and 167: 156 Mathematical Induction To round
Page 168 and 169: 158 Mathematical Induction Proposit
Page 170 and 171: 160 Mathematical Induction Strong i
Page 174 and 175: 164 Mathematical Induction We next
Page 178 and 179: 168 Mathematical Induction 13. For
Page 181: 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
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248 Solutions 7. P ⇒ Q = (P∧
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250 Solutions Section 2.10 Negate t
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252 Solutions (b) How many such cod
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254 Solutions 5. How many 7-digit b
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256 Solutions 15. If n ∈ Z, then
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258 Solutions 7. Proposition Suppos
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260 Solutions 27. If a ≡ 0 (mod 4
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262 Solutions 7. If a, b ∈ Z, the
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264 Solutions This expresses a 3 +
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266 Solutions 19. If n ∈ N, then
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268 Solutions Chapter 8 Exercises 1
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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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