An Event Independent of Itself Must Be Trivial

Let $A$ be an event in a probability space. (a) Write the independence condition $P(A \cap A) = P(A) \cdot P(A)$ and deduce which values of $P(A)$ satisfy it. (b) On $\Omega = \{1,2,3,4\}$ with uniform probability, verify your answer by testing $A = \{1\}$, $A = \{1,2\}$, $A = \emptyset$, and $A = \Omega$. (c) Interpret: what does it mean probabilistically for an event to be independent of itself?