# Identity of Power Set with Union

## Theorem

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

Consider the algebraic structure $\struct {\powerset S, \cup}$, where $\cup$ denotes set union.

Then the empty set $\O$ serves as the identity for $\struct {\powerset S, \cup}$.

## Proof

$\O \in \powerset S$

From Union with Empty Set:

$\forall A \subseteq S: A \cup \O = A = \O \cup A$

By definition of power set:

$A \subseteq S \iff A \in \powerset S$

So:

$\forall A \in \powerset S: A \cup \O = A = \O \cup A$

Thus we see that $\O$ acts as the identity for $\struct {\powerset S, \cup}$.

$\blacksquare$