Definition:Bounded Below

Definition

Ordered Set

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

A subset $T \subseteq S$ is bounded below (in $S$) if:

$\exists m \in S: \forall a \in T: m \preceq a$

That is, there is an element of $S$ (at least one) that precedes all the elements in $T$.

If there is no such element, then $T$ is unbounded below (in $S$).

Mapping

Let $f: S \to T$ be a mapping whose codomain is a poset $\left({T, \preceq}\right)$.

Then $f$ is said to be bounded below (in $T \ $) by the lower bound $L$ iff:

$\forall x \in S: L \preceq f \left({x}\right)$

That is, iff $f \left({S}\right) = \left\{{f \left({x}\right): x \in S}\right\}$ is bounded below by $L$.

If there is no such $L \in T$ then $f$ is unbounded below (in $T \ $).

Also see

• Lower Bound
• Bounded Above
• Upper Bound
• Bounded