The 2-tuples (ordered pairs) of integers is the set Z x Z and can be given an equivalence relation under the condition (a, b)R(c, d) = ad = cb a) Explain why this forms an equivalence relation by using simple numbers in an example. You must explicitly show why this relation satisfies the properties of reflexive, symmetric, and transitive. b) Give examples demonstrating that for two equivalence classes [(a, b)] and [(x,y)], either [(a, b)] = [(x,y)] or [(a, b)] n [(x,y)] = Ø. Give at least one example for each case. Recall that [(a, b)] = {(c, d) E Z × Z: (a, b)R(c, d)}. %3D

The 2-tuples (ordered pairs) of integers is the set Z x Z and can be given an equivalence relation under the condition (a, b)R(c, d) = ad = cb a) Explain why this forms an equivalence relation by using simple numbers in an example. You must explicitly show why this relation satisfies the properties of reflexive, symmetric, and transitive. b) Give examples demonstrating that for two equivalence classes [(a, b)] and [(x,y)], either [(a, b)] = [(x,y)] or [(a, b)] n [(x,y)] = Ø. Give at least one example for each case. Recall that [(a, b)] = {(c, d) E Z × Z: (a, b)R(c, d)}. %3D

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter1: Fundamentals

Section1.7: Relations

Problem 3E: a. Let R be the equivalence relation defined on Z in Example 2, and write out the elements of the...

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