Indepence of Two Events

Question

Suppose X and Y are r.vs and $(X-Y)$ and $Y$ are independent. Does this imply the independence of $X$ and $Y$? Since $P((X-y)|Y=y) = P(X-y)$, it seems $X$ and $Y$ are independent. How do you formally show this?

Answer 1

Let $Y,Z$ be independent, identically distributed random variables, each uniformly distributed on $\{0,1\}$, and let $X=Y+Z$.

Then $X-Y=Z$, so $X-Y,Y$ are independent.

But $X,Y$ are not independent, since for example, $P(X=0)={\large{\frac{1}{4}}}$,$\;$whereas $P(X=0{\,\mid\,}Y=1)=0$.