Power Set is Algebra of Sets

Theorem
Let $S$ be a set.

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

Then $\powerset S$ is an algebra of sets where $S$ is the unit.

Proof
From Power Set is Closed under Intersection and Power Set is Closed under Symmetric Difference, we have that:


 * $(1): \quad \forall A, B \in \powerset S: A \cap B \in \powerset S$
 * $(2): \quad \forall A, B \in \powerset S: A * B \in \powerset S$

From the definition of power set:
 * $\forall A \in \powerset S: A \subseteq S$

and so $S$ is the unit of $\powerset S$.

Thus we see that $\powerset S$ is a ring of sets with a unit.

Hence the result, by definition of an algebra of sets.