Can two discrete random variables $X$ and $Y$ be independent of eachother, if there exist elements $x_0$ and $y_0$ such that $\{X=x_0\}$ and $\{Y=y_0\}$ are the same set (that has nonzero and nonunit probability)?
I'm wrestling with the definition of independence of random variables currently and it seems to me that two such random variables cannot be independent, because to be independent for all $x$ and $y$ the sets $\{X=x\}$ and $\{Y=y\}$ have two be independent (in the sense of independent events). But identical sets $A$ cannot be independent, since $P(A\cap A)=P(A) \neq P(A)\cdot P(A)$ (unless the have probability $0$ or $1$ which I both excluded).