48.9k views
2 votes
Consider f:A→B. Prove that f is injective if and only if X = (f⁻¹) (f(X)) for all X⊆A. Prove that f is surjective if and only if f(f⁻¹) (Y)) = Y for all Y⊆B.

User Zirael
by
8.3k points

1 Answer

3 votes

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.

User Abdul Abdurahim
by
8.1k points
Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.

9.4m questions

12.2m answers

Categories