Power Set is Boolean Ring

Theorem

Let $S$ be a set, and let $\powerset S$ be its power set.

Denote with $\symdif$ and $\cap$ symmetric difference and intersection, respectively.

Then $\struct {S, \symdif, \cap}$ is a Boolean ring.

Proof

From Symmetric Difference with Intersection forms Ring, $\struct {S, \symdif, \cap}$ is a ring with unity.

By Set Intersection is Idempotent, $\cap$ is idempotent.

It follows that $\struct {S, \symdif, \cap}$ is a Boolean ring.

$\blacksquare$

Sources

• 2008: Paul Halmos and Steven Givant: Introduction to Boolean Algebras ... (previous) ... (next): $\S 3$