# Definition:Smallest

## Definition

### Ordered Set

Let $\struct {S, \preceq}$ be an ordered set.

An element $x \in S$ is the smallest element if and only if:

$\forall y \in S: x \preceq y$

That is, $x$ strictly precedes, or is equal to, every element of $S$.

The Smallest Element is Unique, so calling it the smallest element is justified.

The smallest element of $S$ is denoted $\min S$.

For an element to be the smallest element, all $y \in S$ must be comparable to $x$.

### Smallest Set

Let $S$ be a set.

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

Let $\mathcal T \subseteq \mathcal P \left({S}\right)$ be a subset of $\mathcal P \left({S}\right)$.

Let $\left({\mathcal T, \subseteq}\right)$ be the ordered set formed on $\mathcal T$ by $\subseteq$ considered as an ordering.

Then $T \in \mathcal T$ is the smallest set of $\mathcal T$ if and only if $T$ is the smallest element of $\left({\mathcal T, \subseteq}\right)$.

That is:

$\forall X \in \mathcal T: T \subseteq X$