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 $B$.

Proof

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

Then by definition of power set:

$A, B \subseteq S$

Then:

\(\ds A \symdif B\)

\(\subseteq\)

\(\ds A \cup B\)

Symmetric Difference is Subset of Union

\(\ds \)

\(\subseteq\)

\(\ds S\)

Union is Smallest Superset

\(\ds \leadsto \ \ \)

\(\ds A \symdif B\)

\(\in\)

\(\ds \powerset S\)

Definition of Power Set

$\blacksquare$

Sources

• 1964: Steven A. Gaal: Point Set Topology ... (previous) ... (next): Introduction to Set Theory: $1$. Elementary Operations on Sets