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l4_interesting - Math 3040: Solutions for Homework 8. (1)...

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Unformatted text preview: Math 3040: Solutions for Homework 8. (1) For each of the following relations on the set of integers Z = { , 1 ,- 1 , 2 ,- 2 , 3 ... } determine: is it reflexive? Is it symmetric? Is it transitive? is it an equiv- alence relation? Prove your claims. (a) ( a,b ) ∈ R iff- 1 ≤ a- b ≤ 1. (b) Let n be a positive integer. ( a,b ) ∈≡ n iff n divides a- b . (c) ( a,b ) ∈ T iff ab is negative. (d) ( a,b ) ∈ S iff ab is nonnegative. Solution. (a) • Reflexive. Let a ∈ Z . Then a- a = 0, so- 1 < a- a < 1. Therefore, aRa . • Symmetric. Let aRb . Then- 1 < a- b < 1, and so multiplying all three sides of the inequality by- 1 gives- 1 < b- a < 1. Therefore bRa . • NOT Transitive. Counterexample: a = 0, b = 1, c = 2. 0- 1 =- 1 and 1- 2 =- 1, so aRb and bRc . However, 0- 2 =- 2, and so ( a,c ) 6∈ R . • R is NOT an equivalence relation. (b) • Reflexive. Let a ∈ Z . Then a- a = 0, so n | 0. Therefore, a ≡ n a . • Symmetric. Let a ≡ n b . Then n | a- b , which means there is an m ∈ Z such that mn = a- b . Then (- m ) n = b- a , and so n | b- a , and thus b ≡ n a . • Transitive. Let a ≡ n b and b ≡ n c . Then there are p,q ∈ Z such that pn = a- b and qn = b- c . Then ( p + q ) n = a- b + b- c = a- c , and so n | a- c , and thus a ≡ n...
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This note was uploaded on 01/25/2011 for the course MATH 3040 taught by Professor Kahn during the Spring '08 term at Cornell.

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l4_interesting - Math 3040: Solutions for Homework 8. (1)...

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