# Zero of Power Set with Intersection

## Theorem

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

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

Then the empty set $\O$ serves as the zero element for $\struct {\powerset S, \cap}$.

## Proof

$\O \in \powerset S$
$\forall A \subseteq S: A \cap \O = \O = \O \cap A$

By definition of power set:

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

So:

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

Thus we see that $\O$ acts as the zero element for $\struct {\powerset S, \cap}$.

$\blacksquare$