Set Rings

Definition

A set ring (or Boolean set ring, ring of sets, Boolean ring of sets) is a nonempty set of sets $R$ such that if

\[ E \in R \quad \text{and} \quad F \in R \]

then

\[ E \cup F \in R \quad \text{and} \quad E \setminus F \in R. \]

In other words, a ring is a nonempty set of sets which is closed under unions and differences.

Every ring contains the empty set, for if $E \in R$, then $E - E = \varnothing \in R$.

Also, since

\[ E \setminus F = (E \cup F) \setminus F, \]

every nonempty set that is closed under unions and proper differences is a ring.