Prove that a function f : A - B is one to one if and only if any non-empty subset S ⊆ A, f^-1 (f(S)) = S. DO NOT assume a priori that the inverse function f^-1 exists; in this question f^-1 (S) denotes the pre-image of S.

Question

Prove that a function f : A -> B is one to one if and only if any non-empty subset S ⊆ A, f^-1 (f(S)) = S. DO NOT assume a priori that the inverse function f^-1 exists; in this question f^-1 (S) denotes the pre-image of S.