Symmetric Difference with Intersection forms Boolean Ring
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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