# Empty Set and Set form Algebra of Sets

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## 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 Set 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.

$\blacksquare$