Given four variables $x, y, z, w$ such that $x$ is independent of $y$, and $z$ is independent of $w$. In general, do we have $x+z$ is independent of $y+w$? My instructor says it is true in class yesterday. However, I keep thinking of counterexamples. Is there any counterexamples to prove it is false anyway?
Update(additional assumption)
Thanks to @LordSharktheUnknown, this statement is not true for sure. However, what if I add an additional assumption that $x<y<z<w$? In this case, the original statement is true. Am I correct this time?