Ring of Sets is Commutative Ring
Jump to navigation
Jump to search
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 $\O$.
$\blacksquare$