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
Contents
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A , B ∈ R impliesA ∩ B ∈ R and -
A , B ∈ R impliesA ∪ B ∈ R .
In measure theory, a ring of sets
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A , B ∈ R impliesA ∖ B ∈ R and -
A , B ∈ R impliesA ∪ B ∈ R .
This implies the empty set is in
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A △ B = ( A ∖ B ) ∪ ( B ∖ A ) and -
A ∩ B = A ∖ ( A ∖ B ) .
(So a ring in the second, measure theory, sense is also a ring in the first, order theory, sense.) Together, these operations give
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A ∪ B = ( A △ B ) △ ( A ∩ B ) and -
A ∖ B = A △ ( A ∩ B ) .
Examples
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 intervals of the form (a, b], a,b in R is a ring in the measure theory 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.
Related structures
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. Conversely, every distributive lattice is isomorphic to a ring of sets; in the case of finite distributive lattices, this is Birkhoff's representation theorem and the sets may be taken as the lower sets of a partially ordered set.
A field of subsets of X is a ring that contains X and is closed under relative complement. Every field, and so also every σ-algebra, is a ring of sets in the measure theory sense.
A semi-ring (of sets) is a family of sets
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∅ ∈ S , -
A , B ∈ S impliesA ∩ B ∈ S , and -
A , B ∈ S impliesA ∖ B = ⋃ i = 1 n C i C 1 , … , C n ∈ S .
Clearly, every ring (in the measure theory sense) is a semi-ring.
A semi-field of subsets of X is a semi-ring that contains X.