I have the following problem : If $X \perp Y$ and $(X \perp Y )| Z$ , is that true that $X \perp Z$ or $Y \perp Z$ if $X$ and $Y$ can take 3 values but $Z$ only takes two.
$X \perp Y $ means $X$ and $Y$ are independent and $(X \perp Y )| Z$ means that $X$ and $Y$ are independent given $Z$.
I found a answer that shows that it is true when $X$ , $Y$ and $Z$ are binary variables, but I'm struggling when $X$ and $Y$ have not only 2 possibilities.
can someone help me finding a counter exemple or a glimpse of proof ?
thank you a lot.