Q. 45.0( 1 Vote )

Let n be a fixed

Answer :

R = {(a, b) : a – b is divisible by n} on Z.

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 × n


a – a is divisible by n


(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) R


a – b = np For some p Z


b – a = n(–p)


b – a is divisible by n


(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) R and (b, c) R


a – b = np and b – c = nq For some p, q Z


a – c = n (p + q)


a – c = is divisible by n


(a, c) R


R is transitive


R being reflexive, symmetric and transitive on Z.


R is an equivalence relation on Z


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