I'm having some difficulty with these two problems,any help is appreciated.

1)Prove that $\displaystyle A \cup \bigcap B=\bigcap \{A\cup X \mid X\in B\}$ given $\displaystyle B$ is not empty

Since B is not empty, let $\displaystyle X\in B$, then it follows $\displaystyle X \in \bigcap B$, so LHS=$\displaystyle A \cup X$ such that $\displaystyle X \in B$. This doesn't look right, so anyone can help me figure out what I did wrong here.

2) Is it true that $\displaystyle A \cup \bigcup B= \bigcup \{A\cup X \mid X \in B\}$? When does equality hold?

Here, can I take the empty set to be B, and the equality will be false because the empty set doesn't contain any element, but we need $\displaystyle X \in B$. Is my argument correct? I can't find condition such that equality holds.