Definition:Smallest

Definition

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 with $x$.

Let $S$ be a set.

Let $\powerset S$ be the power set of $S$.

Let $\TT \subseteq \powerset S$ be a subset of $\powerset S$.

Let $\struct {\TT, \subseteq}$ be the ordered set formed on $\TT$ by the subset relation.

Then $T \in \TT$ is the smallest set of $\TT$ if and only if $T$ is the smallest element of $\struct {\TT, \subseteq}$.

That is:

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