Set System Closed under Symmetric Difference is Abelian Group
Theorem
Let $\SS$ be a system of sets.
Let $\SS$ be such that:
- $\forall A, B \in \SS: A \symdif B \in \SS$
where $A \symdif B$ denotes the symmetric difference between $A$ and $B$.
Then $\struct {\SS, \symdif}$ is an abelian group.
Proof
Group Axiom $\text G 0$: Closure
By presupposition on $\SS$, $\struct {\SS, \symdif}$ is closed.
$\Box$
Group Axiom $\text G 1$: Associativity
- $\forall A, B, C \in \SS: \paren {A \symdif B} \symdif C = A \symdif \paren {B \symdif C}$ as Symmetric Difference is Associative.
So $\symdif$ is associative.
$\Box$
Group Axiom $\text G 2$: Existence of Identity Element
From Symmetric Difference with Self is Empty Set, we have that:
- $\forall A \in \SS: A \symdif A = \O$
So it is clear that $\O$ is in $\SS$, from the fact that $\struct {\SS, \symdif}$ is closed.
Then we have:
- $\forall A \in \SS: A \symdif \O = A = \O \symdif A$ from Symmetric Difference with Empty Set and Symmetric Difference is Commutative.
Thus $\O$ acts as an identity.
$\Box$
Group Axiom $\text G 3$: Existence of Inverse Element
From the above, we know that $\O$ is the identity element of $\struct {\SS, \symdif}$.
We also noted that
- $\forall A \in \SS: A \symdif A = \O$
From Symmetric Difference with Self is Empty Set.
Thus each $A \in \SS$ is self-inverse.
$\Box$
Commutativity
- $\forall A, B \in \SS: A \symdif B = B \symdif A$ as Symmetric Difference is Commutative.
So $\symdif$ is commutative.
$\Box$
We see that $\struct {\SS, \symdif}$ is closed, associative, commutative, has an identity element $\O$, and each element has an inverse (itself), so it satisfies the criteria for being an abelian group.
$\blacksquare$