Empty Set and Set form Algebra of Sets

Theorem
Let $S$ be any non-empty set.

Then $\set {S, \O}$ is (trivially) an algebra of sets, where $S$ is the unit.

Proof
From Union is Idempotent:
 * $S \cup S = S$

and
 * $\O \cup \O = \O$

Then from Union with Empty Set:
 * $S \cup \O = S$

So $\set {S, \O}$ is closed under union.

From Relative Complement of Empty Set:
 * $\relcomp S \O = S$

and from Relative Complement with Self is Empty Set:
 * $\relcomp S S = \O$

so $\set {S, \O}$ is closed under complement.

Hence the result, by definition of algebra of sets.