Final answer:
In order to prove that f is injective if and only if X = (f-1) (f(X)) for all X⊆A, we need to prove both directions of the statement. The proof for f being surjective if and only if f(f-1) (Y)) = Y for all Y⊆B is similar, but with the concepts of surjectivity and inverse images.
Step-by-step explanation:
In order to prove that f is injective if and only if X = (f-1) (f(X)) for all X⊆A, we need to prove both directions of the statement.
Direction 1: If f is injective, then X = (f-1) (f(X)) for all X⊆A
Let's assume that f is injective. To prove X = (f-1) (f(X)) for all X⊆A, we can start by proving the inclusion in one direction. Suppose that X is a subset of A and consider an arbitrary element x in X. Since f is injective, there exists a unique element y in B such that f(x) = y. Now, let z be an element in (f-1) (f(X)). This means that there is an element w in X such that f(w) = z. Since f(x) = y and f(w) = z, we have y = z. Therefore, z is in X. Hence, (f-1) (f(X)) is a subset of X.
To prove the other inclusion, let z be an element in X. This means that there exists an element w in X such that f(w) = z. By definition of inverse image, we have w in (f-1) (f(X)). Therefore, X is a subset of (f-1) (f(X)).
Since we have shown both inclusions, we can conclude that X = (f-1) (f(X)) for all X⊆A if f is injective.
Direction 2: If X = (f-1) (f(X)) for all X⊆A, then f is injective
Now, let's assume that X = (f-1) (f(X)) for all X⊆A. Our goal is to prove that f is injective. Suppose that f is not injective, which means that there exist distinct elements x and y in A such that f(x) = f(y). Let's consider the set X = {x}. Since f is not injective, there must exist an element z in A such that f(z) = f(x). However, since f(x) = f(y), we also have f(z) = f(y). Therefore, x and y are both in (f-1) (f(X)). But this means that x = y, which contradicts our assumption that x and y are distinct. Hence, our assumption that f is not injective must be false, and we can conclude that f is injective if X = (f-1) (f(X)) for all X⊆A.
The proof for f being surjective if and only if f(f-1) (Y)) = Y for all Y⊆B is similar, but with the concepts of surjectivity and inverse images.