Discreete math functions

Question

Let A, B be sets and f : A → B a function between them. Let also C, D ⊆ A and E, F ⊆ B be subsets. Prove the following.

1. f(C ∩ D) ⊆ f(C) ∩ f(D)

2. f⁻¹(E ∩ F) = f⁻¹(E) ∩ f⁻¹(F)

3. f⁻¹(C) \ f⁻¹(D) = f⁻¹(C \ D)

Answer 1

For 1:

For this one you take an element x∈f(C∩D). You want to prove that this element is also in f(C)∩f(D). That is, you want to prove that x∈f(C) and also x∈f(D). Now since x∈f(C∩D) you know that there is some y∈C∩D such that x=f(y). Now y∈C∩D so in particular y∈C, so that means x=f(y)∈f(C). The same argument works to show that x∈f(D).

Hence in all x is both an element of f(C) and f(D), so x∈f(C)∩f(D).

The rest follow a similar procedure.