Ring of sets
Family closed under unions and relative complements / From Wikipedia, the free encyclopedia
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In mathematics, there are two different notions of a ring of sets, both referring to certain families of sets.
In order theory, a nonempty family of sets is called a ring (of sets) if it is closed under union and intersection.[1] That is, the following two statements are true for all sets and ,
- implies and
- implies
In measure theory, a nonempty family of sets is called a ring (of sets) if it is closed under union and relative complement (set-theoretic difference).[2] That is, the following two statements are true for all sets and ,
- implies and
- implies
This implies that a ring in the measure-theoretic sense always contains the empty set. Furthermore, for all sets A and B,
which shows that a family of sets closed under relative complement is also closed under intersection, so that a ring in the measure-theoretic sense is also a ring in the order-theoretic sense.