Proofs a logical form

Question

Let $\mathcal F$ be a non-empty family of sets with $A\in\mathcal F$.

$(a)$ Prove $A\subset\bigcup\mathcal F$

$(b)$ Prove $\cap\mathcal F\subset A$

$(c)$ Why was the assumption that $\mathcal F$ is nonempty needed?

Was it needed for both parts $(a)$ and $(b)$, or just one? If just one, which one?

Answer 1

a) Prove ⊂⋃ℱ

I did Let x ∈ A then x ∈ F since A ∈ F. Since x ∈ F, it follows that x ⊂ UF. Since x ∈ A then A ⊂ UF.

Is it correct?