Ring of Sets is Commutative Ring

Theorem

A ring of sets $\struct {\RR, \symdif, \cap}$ is a commutative ring whose zero is $\O$.

Proof

By definition, the operations $\cap$ and $\symdif$ are closed in $\RR$.

Hence we can apply the following results:

• Set System Closed under Symmetric Difference is Abelian Group: $\struct {\RR, \symdif}$ is an abelian group.

• Set System Closed under Intersection is Commutative Semigroup: $\struct {\RR, \cap}$ is a commutative semigroup.

• Intersection Distributes over Symmetric Difference.

• The identity of $\struct {\RR, \symdif}$ is $\O$, and this, by definition, is the zero.

So $\struct {\RR, \symdif, \cap}$ is a commutative ring whose zero is $\O$.

$\blacksquare$