Suppose that $X$ and $Y$ are independent random variables. Is $X + Y$ independent from $X$?
Intuitively I think it should be so - because if information about $X$ gave information about $X + Y$, it should also give information about $Y$ by subtracting X - which then rephrases as, if $X$ and $Y$ are not independent, are $X + Y$ and $Y$ dependent? To answer this I would need to say something about $\sigma(X + Y)$ (in terms of $\sigma(X)$ and $\sigma(Y))$, but I am not sure how to do so.
Or is it false? I don't know.
Edit: Duplicate of Are $X$ and $X+Y$ independent, if $X$ and $Y$ are independent?