# Definition:Smallest/Set

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## Definition

### Definition 1

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:

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

### Definition 2

Let $\mathcal A$ be a set of sets or a class of sets.

Then a set $M$ is the least element of $\mathcal A$ (with respect to the subset relation) if and only if:

$(1): \quad M \in \mathcal A$
$(2): \quad \forall S: \left({S \in \mathcal A \implies M \subseteq S}\right)$