"Minimal Cover" in Finite Collection of Sets?

[WWGD]

Hi All,
Say we have a finite collection $S_1,...,S_n$ of sets , which are not all pairwise disjoint , and we want
to find the minimal collection of the $S_j$ whose union is $\cup S_j$ . Is there
any theorem, result to this effect?

I would imagine that making the $S_j$ pairwise-disjoint would help. Is there some other way?
I think I remember some results about results elated to minimal systems of representatives, maybe would also work?

 

[mfb]

An interesting problem!
I don't have a full solution but here is an intermediate step:

Without loss of generality, you can assume that $S_i \subset \cup S_{j\neq i}$ for all i. If this would be wrong, you know S_i has to be part of the minimal cover, this allows to restrict the problem to the subproblem without that S_i and its elements. This step can be repeated until the first condition is true, or the remaining collection of sets is empty.

Also, for each pair of sets S_i, S_j, $S_i \not\subset S_j$ otherwise you can remove S_i.