Ring of sets

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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 - 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 \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.

[edit] See also

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