Empty Set and Set form Algebra of Sets

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

Then $\left\{{S, \varnothing}\right\}$ is (trivially) an algebra of sets, where $S$ is the unit.

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

and
 * $\varnothing \cap \varnothing = \varnothing$

Then from Union with Empty Set:
 * $S \cap \varnothing = S$

So $\left\{{S, \varnothing}\right\}$ is closed under $\cap$.

From Relative Complement of Empty Set:
 * $\complement_S \left({\varnothing}\right) = S$

and from Relative Complement with Self is Empty Set:
 * $\complement_S \left({S}\right) = \varnothing$

so $\left\{{S, \varnothing}\right\}$ is closed under complement.

Hence the result, by definition of algebra of sets.