Ring of Sets is Commutative Ring

Theorem

A ring of sets $\left({\mathcal R, *, \cap}\right)$ is a commutative ring whose zero is $\varnothing$.

Proof

By definition, the operations $\cap$ and $*$ are closed in $\mathcal R$.

Hence we can apply the following results:

• The identity of $\left({\mathcal R, *}\right)$ is $\varnothing$, and this, by definition, is the zero.

So $\left({\mathcal R, *, \cap}\right)$ is a commutative ring whose zero is $\varnothing$.

$\blacksquare$