# Conditionally independent and intersection

I'm trying to show that, given events $A,B,C,D$, such that $A,B$ are conditionally independent given $C$, whether or not $A,B$ are conditionally independent given $C\cap D$.

I spent a couple of hours trying to figure out whether it was true or not, but haven't made significant progress. Can anyone give me some hints?

Thanks for helping!

Perhaps the quickest way to see that $A$ and $B$ aren't in general conditionally independent given $C\cap D$ is to take the entire space for $C$. Then $A$ and $B$ being conditionally independent given $C$ is equivalent to $A$ and $B$ being independent, and $C\cap D=D$. Certainly not all independent events are conditionally independent given arbitrary $D$.