Q. 64.0( 4 Votes )

m is said to be related to n if m and n are integers and m – n is divisible by 13. Does this define an equivalence relation?

Answer :

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

Reflexivity : For Reflexivity, we need to prove that-


(a, a) R


Let m Z


m – m = 0


m – m is divisible by 13


(m, m) R


R is reflexive


Symmetric : For Symmetric, we need to prove that-


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


Let m, n Z and (m, n) R


m – n = 13p For some p Z


n – m = 13 × (–p)


n – m is divisible by 13


(n – m) 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 (m, n) R and (n, q) R For some m, n, q Z


m – n = 13p and n – q = 13s For some p, s Z


m – q = 13 (p + s)


m – q is divisible by 13


(m, q) R


R is transitive


Hence, R is an equivalence relation on Z.


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