Independence Is Not Closed Under Unions

Let $A$, $B$, $C$ be events with $A \perp\!\!\perp B$ and $A \perp\!\!\perp C$. (a) Prove that if $A$, $B$, $C$ are mutually independent, then $A \perp\!\!\perp (B \cup C)$. (b) Now let $\Omega = \{1,2,3,4\}$ with uniform probability, $A = \{1,2\}$, $B = \{1,3\}$, $C = \{1,4\}$. Verify that $A \perp\!\!\perp B$ and $A \perp\!\!\perp C$. (c) Show that $A$ and $B \cup C$ are not independent. Why doesn't pairwise independence suffice?