Ring of sets

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 ${\displaystyle {\mathcal {R}}}$ is called a ring (of sets) if it is closed under union and intersection.^[1] That is, the following two statements are true for all sets ${\displaystyle A}$ and ${\displaystyle B}$,

1. ${\displaystyle A,B\in {\mathcal {R}}}$ implies ${\displaystyle A\cup B\in {\mathcal {R}}}$ and
2. ${\displaystyle A,B\in {\mathcal {R}}}$ implies ${\displaystyle A\cap B\in {\mathcal {R}}.}$

In measure theory, a nonempty family of sets ${\displaystyle {\mathcal {R}}}$ is called a ring (of sets) if it is closed under union and relative complement (set-theoretic difference).^[2] That is, the following two statements are true for all sets ${\displaystyle A}$ and ${\displaystyle B}$,

1. ${\displaystyle A,B\in {\mathcal {R}}}$ implies ${\displaystyle A\cup B\in {\mathcal {R}}}$ and
2. ${\displaystyle A,B\in {\mathcal {R}}}$ implies ${\displaystyle A\setminus B\in {\mathcal {R}}.}$

This implies that a ring in the measure-theoretic sense always contains the empty set. Furthermore, for all sets A and B,

${\displaystyle A\cap B=A\setminus (A\setminus B),}$

which shows that a family of sets closed under relative complement is also closed under intersection, so that a ring in the measure-theoretic sense is also a ring in the order-theoretic sense.