# "Minimal Cover" in Finite Collection of Sets?

1. May 26, 2015

### 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?

Last edited: May 26, 2015
2. May 28, 2015

### Staff: Mentor

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 Si has to be part of the minimal cover, this allows to restrict the problem to the subproblem without that Si 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 Si, Sj, $S_i \not\subset S_j$ otherwise you can remove Si.