XOR Pairwise Independence Without Mutual Independence

Let $X$ and $Y$ be independent $\text{Bernoulli}(1/2)$ random variables. Define $Z = X \oplus Y$ (where $\oplus$ denotes addition modulo 2). (a) Show that $Z \sim \text{Bernoulli}(1/2)$. (b) Show that any two of $\{X, Y, Z\}$ are independent. (c) Are $X$, $Y$, $Z$ mutually independent? Exhibit a specific joint event that violates the mutual independence condition.