Given $X_1+X_2=0$, exactly one of the two is $+1$

Let $X_1$ and $X_2$ be two iid random variables with $P(X_1=1)=P(X_1=-1)=\tfrac12$. Let $Z=X_1+X_2$, $A_i=X_i^{-1}(\{1\})$, $i=1,2$. Which of the following is not true on $Z^{-1}(\{0\})$?