Symmetric Difference with Intersection forms Boolean Ring

Theorem

Let $S$ be a set.

Let:

$\symdif$ denote the symmetric difference operation

$\cap$ denote the set intersection operation

$\powerset S$ denote the power set of $S$.

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

Proof

From Symmetric Difference with Intersection forms Ring:

$\struct {\powerset S, \symdif, \cap}$ is a commutative ring with unity.

From Set Intersection is Idempotent, $\cap$ is an idempotent operation on $S$.

Hence the result by definition of Boolean ring.

$\blacksquare$

Sources

• 1964: Steven A. Gaal: Point Set Topology ... (previous) ... (next): Introduction to Set Theory: $1$. Elementary Operations on Sets