# Ring of Sets is Commutative Ring

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## 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.

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

$\blacksquare$