Independent family of sets (${\bigcap }_{j=1}^m{A}_j^{{\epsilon }_j}\ne \varnothing$)

A family of subsets of $X$ is independent when every finite choice of distinct members and their complements has nonempty intersection. Write $A^1=A$ and $A^0=X\setminus A$ in the displayed condition. Every assignment of zero or one to the family then gives a collection with the finite intersection property, which can be extended to an ultrafilter.