Symmetric Difference on Power Set forms Abelian Group

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Theorem

Let $S$ be a set such that $\O \subset S$ (that is, $S$ is non-empty).

Let $A * B$ be defined as the symmetric difference between $A$ and $B$.

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


Then the algebraic structure $\struct {\powerset S, *}$ is an abelian group.


Proof

From Power Set is Closed under Symmetric Difference, we have that $\struct {\powerset S, *}$ is closed.


The result follows directly from Set System Closed under Symmetric Difference is Abelian Group.

$\blacksquare$


Sources