Ring of Sets is Commutative Ring

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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 $\varnothing$.

$\blacksquare$