$X \cup ( \bigcap_{A \in \mathscr{A}} A ) = \bigcap_{A \in \mathscr{A} } (X \cup A ) $.
Attempt
We have $x \in X \cup ( \bigcap_{A \in \mathscr{A}} A )$ iff $x \in X$ or $x \in A $ for all $A \in \mathscr{A}$. This is true iff $x \in X \cup A $ for all $A \in \mathscr{A}$ and this holds iff $x \in \bigcap_{A \in \mathscr{A} } (X \cup A )$.
IS this corrrect? or Do I need to show more work? thanks