Uniform distribution- independent random variable max and min

Question

Let $X,Y \sim U_{[-1,1]}$ are independent random variables. Let us define $U=\max (X,Y)$ and $V=\min(X,Y)$. Are random variables $U,V$ independent ?

Answer 1

$P(V > 0) = P(X > 0 \wedge Y > 0) = 0.5 \times 0.5 = 0.25$

However, $P(V > 0 | U = 0) = 0$, since $U = 0 \implies X \leq 0 \wedge Y \leq 0$.

i.e. the distribution of $V | U$ is not the same as the unconditional distribution of $V$, so they are not independent.