Show that the following statements are equivalent for a map

Show that the following statements are equivalent for a map f: X rightarrow Y: f is surjective; f(X) = Y; AC X, f(A^c) = [f(A)]^c.

Solution

Let the function, f : X Y be surjective We are required to prove that f(X) = Y. We already know that f(X) Y if f: X Y is a well-defined function. Now, we have to prove that Y f(X). To prove this, we will use the definition of a subset i.e. we will prove that every element of Y is also an element of f(X). Let y be an arbitrary element of Y. Then, since f is surjective, there exists an element x in X such that y = f(x). Thus, Y f(X). We already know that f(X) Y. Therefore Y = f(X).

If x is an arbitrary element of A^c , then f(x) does not belong to f(A) ( as f(A) is the set of images of elements in A). Therefore, f(x) [f(A)]^c . Thus f(A^c) [f(A)]^c. Now, let y be an arbitrary element of [f(A)]^c. Then y does not belong to f(A). Further, since f(A) is the image of all elements in A, hence y f(A^c). Thus [f(A)]^c f(A^c) . Therefore, [f(A)]^c = f(A^c)