Ring of sets

In mathematics, a nonempty collection of sets $mathcal\{R\}$ is called a ring (of sets) if it is closed under intersection and symmetric difference. That is, for any $A,Binmathcal\{R\}$,
#$A\; cap\; B\; in\; mathcal\{R\}$
#$A\; riangle\; B\; in\; mathcal\{R\}$where $riangle$ represents the symmetric difference :$A\; Delta\; B\; =\; (A\; -\; B)\; cup\; (B\; -\; A).$

A ring of sets forms an algebraic ring (possibly without unit) under these two operations. Intersection distributes over symmetric difference:

$A\; cap\; (B\; riangle\; C)\; =\; (A\; cap\; B)\; riangle\; (A\; cap\; C)$

The empty set is the identity element for $riangle$, and the union of all the sets, if it is in the ring, is the identity element for $cap$, making it a unit ring.

Given any set "X", the power set of "X" forms a discrete ring of sets, while the collection {∅,"X"} constitutes the indiscrete ring of sets. Any field of sets and so also any σ-algebra also is a ring of sets.

See also

* Delta ring
* Sigma ring