Zero-Probability Events Are Independent of Everything

Let $(\Omega, \mathcal{F}, P)$ be a probability space and let $A$ be an event with $P(A) = 0$. (a) Prove that $A$ is independent of every event $B \in \mathcal{F}$. (b) Let $\Omega = \{1,2,3,4,5,6\}$ with uniform probability. Set $A = \emptyset$ and $B = \{1,2,3\}$. Verify the independence condition directly. (c) Does the same conclusion hold when $P(A) = 1$? Prove or give a counterexample.