Example 5, Show that the relation R in the set Z of integers given by R = {(a, b) : 2 divides a b} is an equivalence relation. R = {(a, b) : 2 divides a b} Check reflexive Since a a = 0 & 2 divides 0 , eg: 0 2 = 0 2 divides a a (a, a) R, R is reflexive. Check symmetric If 2 divides a b , then 2 divides (a b) i.e. b a Hence, If (a, b) R, then (b, a) R R is symmetric Check transitive If 2 divides (a b) , & 2 divides (b c) , So, 2 divides (a b) + (b c) also So, 2 divides (a c) If (a, b) R and (b, c) R, then (a, c) R Therefore, R is transitive. Thus, R is an equivalence relation in Z.

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Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.

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