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l4_interesting - Math 3040 Solutions for Homework 8(1 For...

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Math 3040: Solutions for Homework 8. (1) For each of the following relations on the set of integers Z = { 0 , 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 ) 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 c . • ≡ n is an equivalence relation.
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