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184 Relations We close this section with several other examples of equivalence relations on infinite sets. Example 11.12 Let P be the set of all polynomials with real coefficients. Define a relation R on P as follows. Given f (x), g(x) ∈ P, let f (x) R g(x) mean that f (x) and g(x) have the same degree. Thus (x 2 + 3x − 4) R (3x 2 − 2) and (x 3 + 3x 2 − 4) ̸R (3x 2 − 2), for example. It takes just a quick mental check to see that R is an equivalence relation. (Do it.) It’s easy to describe the equivalence classes of R. For example, [3x 2 +2] is the set of all polynomials that have the same degree as 3x 2 + 2, that is, the set of all polynomials of degree 2. We can write this as [3x 2 + 2] = { ax 2 + bx + c : a, b, c ∈ R, a ≠ 0 } . Example 11.8 proved that for a given n ∈ N the relation ≡ (mod n) is reflexive, symmetric and transitive. Thus, in our new parlance, ≡ (mod n) is an equivalence relation on Z. Consider the case n = 3. Let’s find the equivalence classes of the equivalence relation ≡ (mod 3). The equivalence class containing 0 seems like a reasonable place to start. Observe that [0] = { x ∈ Z : x ≡ 0(mod 3) } = { x ∈ Z : 3 | (x − 0) } = { x ∈ Z : 3 | x } = { ...,−3,0,3,6,9,... } . Thus the class [0] consists of all the multiples of 3. (Or, said differently, [0] consists of all integers that have a remainder of 0 when divided by 3). Note that [0] = [3] = [6] = [9], etc. The number 1 does not show up in the set [0] so let’s next look at the equivalence class [1]: [1] = { x ∈ Z : x ≡ 1(mod 3) } = { x ∈ Z : 3 | (x − 1) } = { ...,−5,−2,1,4,7,10,... } . The equivalence class [1] consists of all integers that give a remainder of 1 when divided by 3. The number 2 is in neither of the sets [0] or [1], so we next look at the equivalence class [2]: [2] = { x ∈ Z : x ≡ 2(mod 3) } = { x ∈ Z : 3 | (x − 2) } = { ...,−4,−1,2,5,8,11,... } . The equivalence class [2] consists of all integers that give a remainder of 2 when divided by 3. Observe that any integer is in one of the sets [0], [1] or [2], so we have listed all of the equivalence classes. Thus ≡ (mod 3) has exactly three equivalence classes, as described above. Similarly, you can show that the equivalence relation ≡ (mod n) has n equivalence classes [0],[1],[2],..., [n − 1].
Equivalence Relations 185 Exercises for Section 11.2 1. Let A = { 1,2,3,4,5,6 } , and consider the following equivalence relation on A: R = { (1,1),(2,2),(3,3),(4,4),(5,5),(6,6),(2,3),(3,2),(4,5),(5,4),(4,6),(6,4),(5,6),(6,5) } List the equivalence classes of R. 2. Let A = { a, b, c, d, e } . Suppose R is an equivalence relation on A. Suppose R has two equivalence classes. Also aRd, bRc and eRd. Write out R as a set. 3. Let A = { a, b, c, d, e } . Suppose R is an equivalence relation on A. Suppose R has three equivalence classes. Also aRd and bRc. Write out R as a set. 4. Let A = { a, b, c, d, e } . Suppose R is an equivalence relation on A. Suppose also that aRd and bRc, eRa and cRe. How many equivalence classes does R have? 5. There are two different equivalence relations on the set A = { a, b } . Describe them. Diagrams will suffice. 6. There are five different equivalence relations on the set A = { a, b, c } . Describe them all. Diagrams will suffice. 7. Define a relation R on Z as xR y if and only if 3x − 5y is even. Prove R is an equivalence relation. Describe its equivalence classes. 8. Define a relation R on Z as xR y if and only if x 2 + y 2 is even. Prove R is an equivalence relation. Describe its equivalence classes. 9. Define a relation R on Z as xR y if and only if 4|(x+3y). Prove R is an equivalence relation. Describe its equivalence classes. 10. Suppose R and S are two equivalence relations on a set A. Prove that R ∩ S is also an equivalence relation. (For an example of this, look at Figure 11.2. Observe that for the equivalence relations R 2 , R 3 and R 4 , we have R 2 ∩R 3 = R 4 .) 11. Prove or disprove: If R is an equivalence relation on an infinite set A, then R has infinitely many equivalence classes. 12. Prove or disprove: If R and S are two equivalence relations on a set A, then R ∪ S is also an equivalence relation on A. 13. Suppose R is an equivalence relation on a finite set A, and every equivalence class has the same cardinality m. Find a formula for |R| in terms of m. 14. Suppose R is a reflexive and symmetric relation on a finite set A. Define a relation S on A by declaring xS y if and only if for some n ∈ N there are elements x 1 , x 2 ,..., x n ∈ A satisfying xRx 1 , x 1 Rx 2 , x 2 Rx 3 , x 3 Rx 4 ,..., x n−1 Rx n , and x n R y. Show that S is an equivalence relation and R ⊆ S. Prove that S is the unique smallest equivalence relation on A containing R. 15. Suppose R is an equivalence relation on a set A, with four equivalence classes. How many different equivalence relations S on A are there for which R ⊆ S?
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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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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 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
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
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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
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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
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Page 230 and 231: 220 Cardinality of Sets And saying
Page 232 and 233: 222 Cardinality of Sets Here is a s
Page 234 and 235: 224 Cardinality of Sets Beginning a
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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
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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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