Power Set is Algebra of Sets

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Theorem

Let $S$ be a set.

Let $\mathcal P \left({S}\right)$ be the power set of $S$.


Then $\mathcal P \left({S}\right)$ 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:

$(1): \quad \forall A, B \in \mathcal P \left({S}\right): A \cap B \in \mathcal P \left({S}\right)$
$(2): \quad \forall A, B \in \mathcal P \left({S}\right): A * B \in \mathcal P \left({S}\right)$

From the definition of power set:

$\forall A \in \mathcal P \left({S}\right): A \subseteq S$

and so $S$ is the unit of $\mathcal P \left({S}\right)$.

Thus we see that $\mathcal P \left({S}\right)$ is a ring of sets with a unit.

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

$\blacksquare$