Question regarding stochastic independence

Question

Let $X,Y,Z$ be random variables and pairwise independent, i.e. $X$ independent from $Y$, $Y$ indepedent from $Z$, and $X$ independent from $Z$. I am interested in a rigorous argument, why $ (X,Y) $ is independent from $Z$?

Answer 1

Counterexample: Let $\Omega$ be the sample space consisting of all the permutations of $(1,2,3)$ as well as the three vectors $(1,1,1), (2,2,2)$ and $(3,3,3)$ (total: $3!+3$ elements in $\Omega$). One vector is chosen at random from $\Omega$, denoted by $(X,Y,Z)$. Clearly, each of these is uniform over $\{1,2,3\}$ and they are pairwise independent, but given $(X,Y)$ the value of $Z$ is uniquely determined, so $(X,Y)$ and $Z$ are not independent.