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150 Disproof 9.3 Disproof by Contradiction Contradiction can be a very useful way to disprove a statement. To see how this works, suppose we wish to disprove a statement P. We know that to disprove P, we must prove ∼ P. To prove ∼ P with contradiction, we assume ∼∼ P is true and deduce a contradiction. But since ∼∼ P = P, this boils down to assuming P is true and deducing a contradiction. Here is an outline: How to disprove P with contradiction: Assume P is true, and deduce a contradiction. To illustrate this, let’s revisit Example 9.3 but do the disproof with contradiction. You will notice that the work duplicates much of what we did in Example 9.3, but is it much more streamlined because here we do not have to negate the conjecture. Example 9.5 Disprove the following conjecture. Conjecture: There is a real number x for which x 4 < x < x 2 . Disproof. Suppose for the sake of contradiction that this conjecture is true. Let x be a real number for which x 4 < x < x 2 . Then x is positive, since it is greater than the non-negative number x 4 . Dividing all parts of x 4 < x < x 2 by the positive number x produces x 3 < 1 < x. Now subtract 1 from all parts of x 3 < 1 < x to obtain x 3 − 1 < 0 < x − 1 and reason as follows: x 3 − 1 < 0 < x − 1 (x − 1)(x 2 + x + 1) < 0 < (x − 1) x 2 + x + 1 < 0 < 1 Now we have x 2 + x + 1 < 0, which is a contradiction because x is positive. Thus the conjecture must be false. ■ Exercises for Chapter 9 Each of the following statements is either true or false. If a statement is true, prove it. If a statement is false, disprove it. These exercises are cumulative, covering all topics addressed in Chapters 1–9. 1. If x, y ∈ R, then |x + y| = |x| + |y|. 2. For every natural number n, the integer 2n 2 − 4n + 31 is prime.
Disproof by Contradiction 151 3. If n ∈ Z and n 5 − n is even, then n is even. 4. For every natural number n, the integer n 2 + 17n + 17 is prime. 5. If A, B,C and D are sets, then (A × B) ∪ (C × D) = (A ∪ C) × (B ∪ D). 6. If A, B,C and D are sets, then (A × B) ∩ (C × D) = (A ∩ C) × (B ∩ D). 7. If A, B and C are sets, and A × C = B × C, then A = B. 8. If A, B and C are sets, then A − (B ∪ C) = (A − B) ∪ (A − C). 9. If A and B are sets, then P(A) −P(B) ⊆ P(A − B). 10. If A and B are sets and A ∩ B = , then P(A) −P(B) ⊆ P(A − B). 11. If a, b ∈ N, then a + b < ab. 12. If a, b, c ∈ N and ab, bc and ac all have the same parity, then a, b and c all have the same parity. 13. There exists a set X for which R ⊆ X and ∈ X. 14. If A and B are sets, then P(A) ∩P(B) = P(A ∩ B). 15. Every odd integer is the sum of three odd integers. 16. If A and B are finite sets, then |A ∪ B| = |A| + |B|. 17. For all sets A and B, if A − B = , then B ≠ . 18. If a, b, c ∈ N, then at least one of a − b, a + c and b − c is even. 19. For every r, s ∈ Q with r < s, there is an irrational number u for which r < u < s. 20. There exist prime numbers p and q for which p − q = 1000. 21. There exist prime numbers p and q for which p − q = 97. 22. If p and q are prime numbers for which p < q, then 2p + q 2 is odd. 23. If x, y ∈ R and x 3 < y 3 , then x < y. 24. The inequality 2 x ≥ x + 1 is true for all positive real numbers x. 25. For all a, b, c ∈ Z, if a| bc, then a| b or a| c. 26. Suppose A, B and C are sets. If A = B − C, then B = A ∪ C. 27. The equation x 2 = 2 x has three real solutions. 28. Suppose a, b ∈ Z. If a| b and b | a, then a = b. 29. If x, y ∈ R and |x + y| = |x − y|, then y = 0. 30. There exist integers a and b for which 42a + 7b = 1. 31. No number (other than 1) appears in Pascal’s triangle more than four times. 32. If n, k ∈ N and ( n k ) is a prime number, then k = 1 or k = n − 1. 33. Suppose f (x) = a 0 + a 1 x + a 2 x 2 + ··· + a n x n is a polynomial of degree 1 or greater, and for which each coefficient a i is in N. Then there is an n ∈ N for which the integer f (n) is not prime. 34. If X ⊆ A ∪ B, then X ⊆ A or X ⊆ B.
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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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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
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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 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
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
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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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