Not the neatest and clearest way to ask this question: what is i, for one? Of course, i is a natural number (i.e., element in N) and etc.

Anyway, supose A =/= B ==> there's an element in A that doesn't belong to B or the other way around, so wrg let us assume there's an element x in A that doesn't belong to B ==> f(A) in the x-th decimal digit will have a 1, whereas f(B) in the x-th decimal digit will have a zero ==> f(A) =/= f(B) and we're done.

Tonio