# Power Set is Closed under Symmetric Difference

## Theorem

Let $S$ be a set.

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

Then:

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

where $A * B$ is the symmetric difference between $A$ and $B$.

## Proof

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

Then by definition of power set:

$A, B \subseteq S$

Then:

 $\displaystyle A * B$ $\subseteq$ $\displaystyle A \cup B$ Symmetric Difference is Subset of Union $\displaystyle$ $\subseteq$ $\displaystyle S$ Union is Smallest Superset $\displaystyle \leadsto \ \$ $\displaystyle A * B$ $\in$ $\displaystyle \powerset S$ Definition of Power Set

$\blacksquare$