# 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:

• 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 $\O$.

$\blacksquare$