Question

Asked May 8, 2019

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.

Step 1

Let *F *be a non trivial on X.

To prove that the statements are equivalent:

Step 2

Proof:

Assume (1) is true.

Assume there is a set *A* that is a subset of *X*.

Then by definition of ultrafilter, for any

Step 3

When *A *belongs to *F*, then *X – A *does not belong to *F* but be...

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