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HW 6 - MATH 290 HOMEWORK 6 SOLUTIONS 1 Dene the relation R...

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MATH 290 – HOMEWORK 6 – SOLUTIONS 1. Define the relation R on the integers Z by xRy if and only if either x | y or y | x . (a) Prove that the relation R is not transitive. (Hint: To show it is not transitive, you must find three integers x , y , and z such that xRy and yRz but x 6 Rz .) (b) For each n Z , define A n to be the set of integers related to n , that is, A n = { x Z : xRn } . Since R is symmetric we also have A n = { x Z : nRx } . Describe explicitly the sets A 20 and A 0 . This description can be done by listing the set, or by giving a precise description of the set in words. (c) Prove that \ n Z A n = {- 1 , 0 , 1 } . (Hint: To prove the inclusion “ ” you might want to try proof by contrapositive, that is, show that if m / ∈ {- 1 , 0 , 1 } then m / T n Z A n .) Solution: (a). Let x = 5, y = 10, and z = 2. Since 5 | 10, xRy and since 2 | 10, yRz . But since 2 does not divide 5 and 5 does not divide 2, 2 6 R 5 and hence R is not transitive. (b). The set A 20 consists of all multiples of 20, that is, all m Z such that 20 | m , together with the factors of 20 and their negatives, that is, all m Z such that m | 20.
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