a mapping is one-to-one if for every pair of subsets A and B of S.

The contrapositive of this is:

if for every pair of subsets A and B of S then is not one to one.

intuitively,

So implies that there is some element in that is not in . So there is an element a in A and an element b in B such that but so so is not one-to-one.

To go the other way, we know that the function is not one-to-one so for some . using A={a} and B = {b}, .