Definition:Smallest

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Definition

Let $\left({S, \preceq}\right)$ be a poset.

An element $x \in S$ is the smallest element iff:

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

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

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

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

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 poset formed on $\mathcal T$ by $\subseteq$ considered as an ordering.

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

That is:

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