Q. 15.0( 8 Votes )

Show that the relation R defined by R = {(a, b): a – b is divisible by 3; a, b Z} is an equivalence relation.

Answer :

We have,

R = {(a,b) : a–b is divisible by 3; a, b Z}


To prove : R is an equivalence relation


Proof :


To prove that relation is equivalence, we need to prove that it is reflexive, symmetric and transitive.


Reflexivity : For Reflexivity, we need to prove that-


(a, a) R


Let a Z


a – a = 0


a – a is divisible by 3


( 0 is divisible by 3).


(a, a) R


R is reflexive


Symmetric : For Symmetric, we need to prove that-


If (a, b) R, then (b, a) R


Let a, b Z and (a, b) R


a – b is divisible by 3


a – b = 3p(say) For some p Z


–( a – b) = –3p


b – a = 3 × (–p)


b – a R


R is symmetric


Transitive : : For Transitivity, we need to prove that-


If (a, b) R and (b, c) R, then (a, c) R


Let a, b, c Z and such that (a, b) R and (b, c) R


a – b = 3p(say) and b – c = 3q(say) For some p, q Z


a – c = 3 (p + q)


a – c = 3 (p + q)


(a, c) R


R is transitive


Since, R is reflexive, symmetric and transitive


R is an equivalence relation.


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