Definition:Lower Bound of Set

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This page is about lower bounds of ordered sets which are bounded below. For other uses, see Definition:Lower Bound.

Definition

Let $\left({S, \preceq}\right)$ be an ordered set.

Let $T$ be a subset of $S$.


A lower bound for $T$ (in $S$) is an element $m \in S$ such that:

$\forall t \in T: m \preceq t$

That is, $m$ precedes every element of $T$.


Subset of Real Numbers

The concept is usually encountered where $\left({S, \preceq}\right)$ is the set of real numbers under the usual ordering $\left({\R, \le}\right)$:


Let $\R$ be the set of real numbers.

Let $T$ be a subset of $S$.


A lower bound for $T$ (in $\R$) is an element $m \in \R$ such that:

$\forall t \in T: m \le t$

That is, $M$ is less than every element of $T$.


Also defined as

Some sources use the terminology the lower bound for the notion of infimum.


Also see


Sources