I am asked to prove the following claim:
If $A$ and $B$ are separate events then $A$ and $B$ cannot be independent.
Intuitively, it is easy to see that the claim is true, since if $A$ occurs then $B$ must not have occurred (hence knowledge about $A$ gives complete knowledge about $B$, which is very strong form dependence).
Mathematically, $A$ and $B$ are independent if $P(A\cap B) = P(A)P(B)$. However, if $A$ and $B$ are separate, then there is nothing in the intersection of $A$ and $B$. Hence $P(A\cap B)=0$. However, $P(A)P(B)>0$ if $A$ and $B$ are possible events... So I have arrived in a contradiction? Is this sufficient to prove the claim?