Question

What is the correct answer to the following blanks to prove that for all subsets A and B of X, F(A ∩ B) ⊆ F(A) ∩ F(B) given let X and Y be any sets, and let F be any function from X to Y?

Answer 1

In step 2, it's shown that x ∈ A ∩ B, so in step 3, it follows by the definition of set intersection that x ∈ A and x ∈ B.

Then in step 4, by definition of image of an element, both F(x) ∈ F(A) and F(x) ∈ F(B).

The rest of the proof follows.