Independence from Boolean Combinations Requires Mutual Independence

Let $A$, $B$, $C$ be events. (a) Prove that if $A$, $B$, $C$ are mutually independent, then $A$ is independent of $B \cap C^c$. (b) Now let $\Omega = \{1,2,3,4\}$ with uniform probability, $A = \{1,2\}$, $B = \{1,3\}$, $C = \{1,4\}$. Verify that $A$, $B$, $C$ are pairwise independent but not mutually independent. (c) Compute $P(A \cap (B \cap C^c))$ and $P(A) \cdot P(B \cap C^c)$. Does the independence $A \perp\!\!\perp (B \cap C^c)$ hold?