Ring of sets

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. 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). 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. 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. However, in general it will not be closed under differences of sets. The open sets and closed sets of any topological space are closed under both unions and intersections. On the real line R, the family of sets consisting of the empty set and all finite unions of half-open intervals of the form , with a, b ∈ R is a ring in the measure-theoretic sense. If T is any transformation defined on a space, then the sets that are mapped into themselves by T are closed under both unions and intersections. If two rings of sets are both defined on the same elements, then the sets that belong to both rings themselves form a ring of sets. A ring of sets in the order-theoretic sense forms a distributive lattice in which the intersection and union operations correspond to the lattice's meet and join operations, respectively.