is the assumption If X⊥⊥Y | Z and X⊥⊥Y then (X⊥⊥Z or Y ⊥⊥Z) true ?
where X⊥⊥Y means is independent of Y and X⊥⊥Y means that X is independent of Y given Z?
I have been trying to prove or disprove the assumption above , but I neither found a proof or a counter exemple.
so far I got that expression $p(z|x,y) = \frac{p(z|x)p(z|y)}{p(z)}$ , but I don't really know how I can use it to prove or disprove.
now I try to find real table and show that that des not hold every time, but I'm stuck...
can anyone give me some help with that question ? (Z is a binary random variable)