What does filter (set theory) mean?
Definitions for filter (set theory)
fil·ter (set the·o·ry)
This dictionary definitions page includes all the possible meanings, example usage and translations of the word filter (set theory).
Wikipedia
Filter (set theory)
In mathematics, a filter on a set X {\displaystyle X} is a family B {\displaystyle {\mathcal {B}}} of subsets such that: X ∈ B {\displaystyle X\in {\mathcal {B}}} and ∅ ∉ B {\displaystyle \emptyset \notin {\mathcal {B}}} if A ∈ B {\displaystyle A\in {\mathcal {B}}} and B ∈ B {\displaystyle B\in {\mathcal {B}}} , then A ∩ B ∈ B {\displaystyle A\cap B\in {\mathcal {B}}} If A , B ⊂ X , A ∈ B {\displaystyle A,B\subset X,A\in {\mathcal {B}}} , and A ⊂ B {\displaystyle A\subset B} , then B ∈ B {\displaystyle B\in {\mathcal {B}}} A filter on a set may be thought of as representing a "collection of large subsets". Filters appear in order, model theory, set theory, but can also be found in topology, from which they originate. The dual notion of a filter is an ideal. Filters were introduced by Henri Cartan in 1937 and as described in the article dedicated to filters in topology, they were subsequently used by Nicolas Bourbaki in their book Topologie Générale as an alternative to the related notion of a net developed in 1922 by E. H. Moore and Herman L. Smith. Order filters are generalizations of filters from sets to arbitrary partially ordered sets. Specifically, a filter on a set is just a proper order filter in the special case where the partially ordered set consists of the power set ordered by set inclusion.
Numerology
Chaldean Numerology
The numerical value of filter (set theory) in Chaldean Numerology is: 5
Pythagorean Numerology
The numerical value of filter (set theory) in Pythagorean Numerology is: 7
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"filter (set theory)." Definitions.net. STANDS4 LLC, 2024. Web. 31 Dec. 2024. <https://www.definitions.net/definition/filter+%28set+theory%29>.
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