What's the reasoning for checking the independence of

$$\min(X,Y)$$ and $$\max(X,Y)$$ for independent r.v.s $X,Y$?

Is it possible that $\min$ and $\max$ both select the same r.v. in which case they would be dependent? No, because that would mean that $X=Y$, i.e. $X$ and $Y$ would not be independent.

Does the independence regarding functions of independent r.v.s apply here? Yes? In that case $X,Y$ independent $\implies$ $\min(X,Y),\space \max(X,Y)$ independent.