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If X is any set, then the power set of X (the family of all subsets of X) forms a ring of sets in either sense.. If (X, ≤) is a partially ordered set, then its upper sets (the subsets of X with the additional property that if x belongs to an upper set U and x ≤ y, then y must also belong to U) are closed under both intersections and unions.
Ring of sets – Family closed under unions and relative complements; ... This page was last edited on 4 July 2024, at 10:04 (UTC).
For a semi-ring , the set of all finite unions of sets in is the ring generated by : = {: = =,} (One can show that () is equal to the set of all finite disjoint unions of sets in ). A content μ {\displaystyle \mu } defined on a semi-ring S {\displaystyle S} can be extended on the ring generated by S . {\displaystyle S.}
A ring is a set R equipped with two binary operations [a] + (addition) and ⋅ (multiplication) satisfying the following three sets of axioms, called the ring axioms: [1] [2] [3] R is an abelian group under addition, meaning that:
As a corollary, if is a ring of sets, then the smallest monotone class containing it coincides with the 𝜎-ring of .. By invoking this theorem, one can use monotone classes to help verify that a certain collection of subsets is a 𝜎-algebra.
Birkhoff (1937) defined a ring of sets to be a family of sets that is closed under the operations of set unions and set intersections; later, motivated by applications in mathematical psychology, Doignon & Falmagne (1999) called the same structure a quasi-ordinal knowledge space. If the sets in a ring of sets are ordered by inclusion, they form ...