if Y1 and Y2 are independent, does it follow that U and X are independent, where U = f1(Y1, Y2) and X= f2(Y1, Y2) If X, Y are iid rvs, and U and Z are r.v.s that can each be written in terms of X and Y, does that mean that U and Z are independent?

Even more trivial:  Let $U = Z$ be the same function of $X$ and $Y$.  Then $U$ and $Z$ are obviously dependent.  Or let $U = g(Z)$ for some non-degenerate function $g$.
