I have two questions concerning Independence between random variables and vectors. Let $X,Y,Z,U:\Omega\rightarrow\mathbb{R}$ be random variables.
If $X$ is independent of $Y$ and $Z$, is $X$ independent of $(Y,Z)$?
If $X$ and $U$ are independent of $Y$ and $Z$, is $(X,U)$ independent of $(Y,Z)$?
MOTIVATION: I know that, if $X$ is independent of $(Y,Z)$, then $X$ is independent to both $Y$ and $Z$ (because the real maps $(y,z)\mapsto y$ and $(y,z)\mapsto z$ are measurable). I would like to know if the converse holds, and if not, under which conditions it does (for instance, if the random variables have a certain distribution).
I do know that, for 1., if $Y$ and $Z$ are also independent, then $X$ is indeed independent of $(Y,Z)$.