Question1-: (Filter generated by a family of sets). Let S ⊆ ℘(X) be nonempty family of subsets of X, and let BS be the family of all finite intersections of elements of S (note that S ⊆ BS). Show that BS is a basis for a filter. Show that this filter is nontrivial if and only if S has the finite intersection property.1 Moreover, prove that this is the smallest filter containing S. (Note that therefore there exist families that are not contained in any nontrivial filter). Question2-: Let F be a nontrivial on X. Prove that the follow- ing statements are equivalent1. F is an ultrafilter.2. (∀A⊆X)A∈F ↔ X−A∈/F.3. (∀A,B⊆X)A∪B∈F → A∈F or B∈F.
Question1-: (Filter generated by a family of sets). Let S ⊆ ℘(X) be nonempty family of subsets of X, and let BS be the family of all finite intersections of elements of S (note that S ⊆ BS). Show that BS is a basis for a filter. Show that this filter is nontrivial if and only if S has the finite intersection property.1 Moreover, prove that this is the smallest filter containing S. (Note that therefore there exist families that are not contained in any nontrivial filter). Question2-: Let F be a nontrivial on X. Prove that the follow- ing statements are equivalent1. F is an ultrafilter.2. (∀A⊆X)A∈F ↔ X−A∈/F.3. (∀A,B⊆X)A∪B∈F → A∈F or B∈F.
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter4: Eigenvalues And Eigenvectors
Section4.6: Applications And The Perron-frobenius Theorem
Problem 36EQ
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Question1-:
(Filter generated by a family of sets). Let S ⊆ ℘(X) be nonempty family of subsets of X, and let BS be the family of all finite intersections of elements of S (note that S ⊆ BS). Show that BS is a basis for a filter. Show that this filter is nontrivial if and only if S has the finite intersection property.1 Moreover, prove that this is the smallest filter containing S. (Note that therefore there exist families that are not contained in any nontrivial filter).
Question2-:
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.
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