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.