Power Set with Union and Intersection forms Boolean Algebra

Theorem
Let $S$ be a set, and let $\powerset S$ be its power set.

Denote with $\cup$, $\cap$ and $\complement$ the operations of union, intersection and complement on $\powerset S$, respectively.

Then $\struct {\powerset S, \cup, \cap, \complement}$ is a Boolean algebra.

Proof
Taking the criteria for definition 1 of a Boolean algebra in turn:

$(\text {BA} 0):$ Closure
$\powerset S$ is closed under both $\cup$ and $\cap$:
 * Power Set is Closed under Intersection
 * Power Set is Closed under Union
 * Power Set is Closed under Complement

$(\text {BA} 1):$ Commutativity
Both $\cup$ and $\cap$ are commutative from Intersection is Commutative and Union is Commutative.

$(\text {BA} 2):$ Distributivity
Both $\cup$ and $\cap$ distribute over the other, from Union Distributes over Intersection and Intersection Distributes over Union.

$(\text {BA} 3):$ Identity Elements
Both $\cup$ and $\cap$ have identities:

From Power Set with Intersection is Monoid‎, $S$ is the identity for $\cap$.

From Power Set with Union is Monoid, $\O$ is the identity for $\cup$.

$(\text {BA} 4):$ Complements
From Union with Complement:


 * $\forall A \in S: A \cup \map \complement A = S$

which is the identity for $\cap$.

From Intersection with Complement:


 * $\forall A \in S: A \cap \map \complement A = \O$

which is the identity for $\cup$.

All the criteria for a Boolean algebra are therefore fulfilled.