Let $X, Y:\ \Omega \mapsto \{-1,1\}$ be two independent random variables such that
$$P(X=\pm 1)=P(Y=\pm 1)=1/2.$$
Let $Z=XY$. Based on how I have reasoned so far, I can tell that $X$ is independent of $Z$ and $Y$ is also independent of $Z$. However, I am not completely sure. Is the following proof correct?
For all $a,b \in \Omega$, we have
$$P(X=a,Z=b)=P(X=a,Y=ab)=P(X=a)P(Y=ab)=1/4=P(X=a)P(Z=b),$$
and in the same way, we can show the independence of $Y$ and $Z$.