Example 5 - R = {(a, b) : 2 divides a-b} is equivalence relation - To prove relation reflexive/trasitive/symmetric/equivalent

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  1. Chapter 1 Class 12 Relation and Functions
  2. Serial order wise

Transcript

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 is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 8 years. He provides courses for Maths and Science at Teachoo. You can check his NCERT Solutions from Class 6 to 12.