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 $A,B\in\mathcal{R}$,

1. $A \cap B \in \mathcal{R}$
2. $A \triangle B \in \mathcal{R}$

where $\triangle$ represents the symmetric difference

$A \Delta B = (A\backslash B) \cup (B\backslash A).$

A ring of sets forms an algebraic ring (possibly without unit) under these two operations. Intersection distributes over symmetric difference: $A \cap (B \triangle C) = (A \cap B) \triangle (A \cap C)$

The empty set is the identity element for $\triangle$, 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