Power Set is Boolean Ring
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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): $\S 3$