Let R be the set of real numbers. Statement-l: A = {(x, y) ∈ R × R : y-x is an integer} is an equivalence relation on R. Statement-II: B = {(x, y) ∈ R × R : x = αy for some rational number α} is an equivalence relation on R?

x - y is an integer. x - x = 0 is also an integer ∴ A is reflexive. y - x is also an integer. Hence, it is symmetric. x - y and y - z are both integers, then their sum (x - y) + (y - z) = x - z is also an integer. Hence, transitive. If a set is symmetric, transitive and reflexive then it also has equivalence relation. Clearly, A is an equivalence relation. Now B, x/y = α is rational but y/x need not be rational i.e. 0/1 is rational but 1/0 is not rational. Hence, B doesn't show equivalence relationship