Let R be the relation on the set of ordered pairs of positive integers such that ((a, b), (c, d)) E R if and only if ad = bc. Arrange the proof of the given statement in correct order to show that R is an equivalence relation. (Prove the given relation is reflexive first, and then symmetric and transitive.) Rank the options below. Hence, R is reflexive. If ((a, b), (c, d)) E E Rand ((c, d), (e, f)) E ER, then ad = bc and cf=de. Hence, Ris transitive. 60 7 3 Multiplying these equations gives acdf= bcde, and since all these numbers are nonzero, we have af = be, so ((a, b), (e, f)) E E R. 2 If ((a, b), (c, d)) E Є R then ad = bc, which also means that cb = da, so ((c, d), (a, b)) Є Є R. Hence, Ris symmetric. ((a, b), (a, b)) E E R because ab = ba. Since R is reflexive, symmetric, and transitive, it is an equivalence relation. 8 00 5 4

Let R be the relation on the set of ordered pairs of positive integers such that ((a, b), (c, d)) E R if and only if ad = bc. Arrange the proof of the given statement in correct order to show that R is an equivalence relation. (Prove the given relation is reflexive first, and then symmetric and transitive.) Rank the options below. Hence, R is reflexive. If ((a, b), (c, d)) E E Rand ((c, d), (e, f)) E ER, then ad = bc and cf=de. Hence, Ris transitive. 60 7 3 Multiplying these equations gives acdf= bcde, and since all these numbers are nonzero, we have af = be, so ((a, b), (e, f)) E E R. 2 If ((a, b), (c, d)) E Є R then ad = bc, which also means that cb = da, so ((c, d), (a, b)) Є Є R. Hence, Ris symmetric. ((a, b), (a, b)) E E R because ab = ba. Since R is reflexive, symmetric, and transitive, it is an equivalence relation. 8 00 5 4

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter1: Fundamentals

Section1.7: Relations

Problem 6TFE: Label each of the following statements as either true or false. Let R be a relation on a nonempty...

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