Let R be the set of real numbers.
Statement-1 : A = {(x, y) Î R × R : y – x is an integer} is an equivalence relation on R.
Statement-2 : B = {(x, y) Î R × R : x = ay for some rational number a} is an equivalence relation on R.
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Statement - 1 :
(i) x – x is an integer " x Î R so A is reflexive relation.
(ii) y – x Î I Þ x – y Î I so A is symmetric relation.
(iii) y – x Î I and z – y Î I y – x + z – y Î I
Þ z – x Î I so A is transitive relation.
Therefore A is equivalence relation.
Statement - 2 :
(i) x = ax when a = 1 Þ B is reflexive relation
(ii) for x = 0 and y = 2, we have 0 = a(2) for a = 0
But 2 = a(0) for no a
so B is not symmetric so not equivalence.