What is the number of all possible relations/intersections of n sets? If n defines number of sets, what is the number of all possible relations between them? For example, when n = 2:
1) A can intersect with B
2) A and B can be disjoint
3) A can be subset of B
4) B can be subset of A
that leaves us with 4 possible relations. Now with the n = 3 it gets tricker (A can intersect with B, but not C or B can be subsect of C and intersect with A there etc). Wondering if there's any formula that can be made to calculate such possible relations. Working on this problem for last couple of days, read about Venn diagrams, Karnaugh map, but still can't figure that one out. Any help is appreciated!

Disclaimer: Not an answer®
I'd like to think about this problem not as sets, but as elements in a partial order. Suppose all sets are different. Define $\mathscr{P} = \langle\mathscr{P}(\bigcup_n A_n), \subseteq\rangle$ as the partial order generated bay subset relation on all "interesting" sets.
Define the operation $\cap$ on $\mathscr{P}$ as
$$
C = A\cap B \iff C = \sup_\subseteq \{D: D\subseteq A \wedge D\subseteq B\}
$$
which is well defined because of, well, set theory....
The question could be then stated as:
Given the sets $A_n$, let $\mathscr{G}$ the subgraph of $\mathscr{P}$ generated by the $A_n$ and closed under the $\cap$ operation. How many non-isomoprhic graph can we get?
