Let $U$ be a finite non-empty set of size $n$. May $E$ be the inclusion relation $\subseteq$ over $P(U)$.
I've seen the proof using Newton's binomial, that $|E| = 3^n$. I'm interested in a more direct combinatorical proof. I've been given a hint: given $A \subseteq B \subseteq U$, $U$ can be represented as a union of 3 disjoint sets. I think this means $A$, $B-A$, and $U-B$. But I couldn't see how that helps.