Independent Events Covering the Sample Space

Let $A$ and $B$ be independent events with $P(A \cup B) = 1$. (a) Using the independence condition and inclusion-exclusion, prove that $(1 - P(A))(1 - P(B)) = 0$. (b) What does this force about $P(A)$ and $P(B)$? (c) On $\Omega = \{1,2,3,4\}$ with uniform probability, find events $A$ and $B$ that are independent, satisfy $P(A \cup B) = 1$, and have $P(A) = 1$ while $P(B) = 1/2$. Verify all three conditions.