# Power Set is Algebra of Sets

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## 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.

$\blacksquare$