Power Set is Closed under Symmetric Difference

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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

$\blacksquare$


Sources