Assume $U$ and $V$ are uniform$(0,1)$ random variables with the following joint distribution: $$F(u,v)=\left\{ \begin{array}{ll} u, &0\leq u \leq v/2 \leq 1/2\\ v/2, & 0 \leq v/2 \leq u < 1 - v/2 \\ u + v - 1, & 1/2 \leq 1 - v/2 \leq u \leq 1 \end{array} \right.$$ The exercise is to show that $\mathbb{P}(V = 1 - |2U -1|) = 1$ and $\text{Cov}(U,V) = 0$.
It seems a simple exercise, but my probability skills got a bit rusty and the book I'm following does not delve deep into the technicalities. How would we proceed from here?
It seems we are interested in finding $P(Z = 0) = 1 - P(Z > 0)$, where $Z := V - 1 + |2U-1|$, but it's not clear how to proceed.