# Infimum of Power Set

## Theorem

Let $S$ be a set.

Let $\mathcal P \left({S}\right)$ be the power set of $S$.

Let $\left({\mathcal P \left({S}\right), \subseteq}\right)$ be the relational structure defined on $\mathcal P \left({S}\right)$ by the relation $\subseteq$.

(From Subset Relation on Power Set is Partial Ordering, this is an ordered set.)

Then the infimum of $\left({\mathcal P \left({S}\right), \subseteq}\right)$ is the empty set $\varnothing$.

## Proof

Follows directly from Empty Set is Subset of All Sets and the definition of infimum.

$\blacksquare$