Warning: f^(1) is NOT a function (unless f is bijective). You cannot plug elements into f^(1)(V). You can only plug in subsets of the codomain into f^(1).

4. Suppose that f : X Y is a function. For any two different y1, y2 Y , show that f^(1)({y1}) f^(1)({y2}) = . 5. Suppose that f : X Y and g : Y Z are functions. Prove that for any subset V Z that f^(1)(g^(1)(V)) = (gf)^(1)(V).