Question

Prove that the relation R = {(a, b) : (a – b) is divisible by 5}, is all equivalence

relation on the set of Integers I​

Answer 1

R = {(a,b):a - b is divisible by 5}

clearly, R is reflexive as a - a = 0 is divisible by 5

R is symmetric also as if (a - b) is divisible by 5, i.e. a - b = 5k then (b - a) = - (a - b) = -5k will also be divisible by 5

R is transitive as if a - b is divisible by 5 i.e. a - b = 5m and b- c is divisible by 5 i.e. b - c = 5n, then

a - c = (a - b) + (b - c) = 5m + 5n = 5 (m + n) is also divisible by 5.

Thus, R is reflexive, symmetry and transitive. hence R is equivalence relation.