under union and relative complement is also closed under symmetric difference and intersection. Conversely, every family of sets closed under both symmetric difference and intersection is also closed under union and relative complement. This is due to the identities and Symmetric difference and intersection together give a ring in the measure-theoretic sense the structure of a boolean ring. In the measure-theoretic sense, a is a ring closed under unions, and a δ-ring is a ring closed under countable intersections. Explicitly, a σ-ring over is a set such that for any sequence we have Given a set a − also called an − is a ring that contains This definition entails that an algebra is closed under absolute complement A σ-algebra is an algebra that is also closed under countable unions, or equivalently a σ-ring that contains In fact, by de Morgan's laws, a δ-ring that contains is necessarily a σ-algebra as well. Fields of sets, and especially σ-algebras, are central to the modern theory of probability and the definition of measures. A is a family of sets with the properties If (3) holds, then if and only if implies and implies for some disjoint Every ring (in the measure theory sense) is a semi-ring. On the other hand, on is a semi-ring but not a ring, since it is not closed under unions. A or is a collection of subsets of satisfying the semiring properties except with (3) replaced with: If then there exists a finite number of mutually disjoint sets such that This condition is stronger than (3), which can be seen as follows. If is a semialgebra and , then we can write for disjoint . Then: and every since it is closed under intersection, and disjoint since they are contained in the disjoint 's. Moreover the condition is strictly stronger: any that is both a ring and a semialgebra is an algebra, hence any ring that is not an algebra is also not a semialgebra (e.g. the collection of finite sets on an infinite set ). See also References External links Ring of sets at Encyclopedia of Mathematics Families of sets Measure theory