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σ-ring
In mathematics, a nonempty collection of Set (mathematics), sets is called a -ring (pronounced ''sigma-ring'') if it is Closure (mathematics), closed under countable Union (set theory), union and Complement (set theory), relative complementation. Formal definition Let \mathcal be a nonempty Family of sets, collection of sets. Then \mathcal is a -ring if: # Closed under countable Union (set theory), unions: \bigcup_^ A_ \in \mathcal if A_ \in \mathcal for all n \in \N # Closed under Complement (set theory), relative complementation: A \setminus B \in \mathcal if A, B \in \mathcal Properties These two properties imply: \bigcap_^ A_n \in \mathcal whenever A_1, A_2, \ldots are elements of \mathcal. This is because \bigcap_^\infty A_n = A_1 \setminus \bigcup_^\left(A_1 \setminus A_n\right). Every -ring is a Delta-ring, δ-ring but there exist δ-rings that are not -rings. Similar concepts If the first property is weakened to closure under finite union (that is, A \cup B \in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Family Of Sets
In set theory and related branches of mathematics, a family (or collection) can mean, depending upon the context, any of the following: set, indexed set, multiset, or class. A collection F of subsets of a given set S is called a family of subsets of S, or a family of sets over S. More generally, a collection of any sets whatsoever is called a family of sets, set family, or a set system. Additionally, a family of sets may be defined as a function from a set I, known as the index set, to F, in which case the sets of the family are indexed by members of I. In some contexts, a family of sets may be allowed to contain repeated copies of any given member, and in other contexts it may form a proper class. A finite family of subsets of a finite set S is also called a '' hypergraph''. The subject of extremal set theory concerns the largest and smallest examples of families of sets satisfying certain restrictions. Examples The set of all subsets of a given set S is called the pow ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |