Power Set is Closed under Symmetric Difference

Theorem

Let $S$ be a set.

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

Then:

$\forall A, B \in \powerset S: A \symdif B \in \powerset S$

where $A \symdif B$ is the symmetric difference between $A$ and $B$.

Let $A, B \in \powerset S$.

Then by definition of power set:

$A, B \subseteq S$

Then:

$\ds A \symdif B$

$\subseteq$

$\ds A \cup B$

$\ds $

$\subseteq$

$\ds S$

$\ds \leadsto \ \ $

$\ds A \symdif B$

$\in$

$\ds \powerset S$

Definition of Power Set

$\blacksquare$