Minimum subset of a set, whose union is same as original set?

Question

hi,
if you have a Set of Sets called S, e.g.
{{1,2},{2,3},{3,4}}
and want to find the/a minimum subset of S called S', which has the property that
union(S)==union(S')
eg
union({{1,2},{2,3},{3,4}})==union({{1,2},{3,4}})
where union(S) means the union of all elements of S ie
union({{1,2},{2,3},{3,4}}) = {1,2,3,4}
then is there a better way than considering all combinations?
my first instinct that the largest component set would always be in the result was proved wrong by
{{1,2,4,5,7,8},{1,2,3,4},{5,6,7,8}}
I looked on Google, but have not found much - Set Covering sounds like a good name for this but seems related to algebra instead?
thanks,
asjf

Answer 1

We both need to read. S' is a subset of S.
1. Take the largest set in S, add it to S'.
2. Take the next largest set in S, call is A. If A
has any elements that aren't in S', add it to S'
3. if there are more sets in S, repeat step 2.Oh, I see now. You can stop after all the elements in the union are reprsented in S' though. Also, he has stated that this algorithm produces tha incorrect result for {{1,2,4,5,7,8},{1,2,3,4},{5,6,7,8}}
i.e. it gives:
{{1,2,4,5,7,8},{1,2,3,4}}
when he wants:
{{1,2,3,4},{5,6,7,8}}
because they are smaller sets.