I need some help with the following exercise:
I don't know how to start with this exercise. I know the definition of a distributive lattice, and I know some properties, namely:
I wanted to use the first property. Is it true that the intersection equals the infimum, and the union equals the supremum? I would say this is true, since it also holds for a power set, which is a more general case. Now, to me it seems then that it is merely a set-theoretic result that $$ A\cap(B\cup B)=(A\cap B)\cup(A\cap C), $$ where $A,B,C\in\operatorname{Up}(C)$. Is this correct?
What you say is correct, but you need to provide a bit more justification: you need to prove that if $A$ and $B$ are up-sets, then so are $A\cup B$ and $A\cap B$. Once you know that, then the fact that they are the join and meet in the power set implies they are also the join and meet in $\mathrm{Up}(P)$.