Let F be a nontrivial on X. Prove that the follow-
ing statements are equivalent
1. F is an ultrafilter.
2. (∀A⊆X)A∈F ↔ X−A∈/F.
3. (∀A,B⊆X)A∪B∈F → A∈F or B∈F.
Let F be a non trivial on X.
To prove that the statements are equivalent:
Assume (1) is true.
Assume there is a set A that is a subset of X.
Then by definition of ultrafilter, for any
When A belongs to F, then X – A does not belong to F but be...
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