# Reducing 3SAT to a Set Splitting Problem

I want to be able to reduce the 3SAT problem to a flavor of set-splitting problem. Basically, given $$n$$ items and $$m$$ subsets $$S_1, S_2,...,S_m$$ of these items I want a yes or no answer based upon the fact if it is possible to color the items red and blue so that each set $$S_i$$ contains at least one red item and at least one blue item.

I know this this problem is in NP and it is fairly simple to prove (at least that is what I think).

I actually want to prove that the problem is NP hard by reduction from 3SAT. I want to be able to convert an instance of 3SAT into an instance of the aforementioned set-splitting such that a yes instance of 3SAT get mapped to a yes instance of set-splitting and vice versa.

I have a basic intuition about how I can do this: given a 3SAT formula $$f$$ on $$n$$ variables I will create $$2n+1$$ items. One to denote each $$x_i$$ one for each $$\overline{x_i}$$ and an extra item. I am not quite sure how to proceed beyond this basic intuition. This might be a gap in my understanding of the problem. Is this the right way to approach this? If so, how can I use this intuition to actually prove that this is NP hard by way of reduction from 3SAT?

Thank you!