Definition:Lower Bound of Set

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This page is about Lower Bound in the context of Ordered Set. For other uses, see Lower Bound.

Definition

Let $\struct {S, \preceq}$ 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 $\struct {S, \preceq}$ is the set of real numbers under the usual ordering $\struct {\R, \le}$:


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:

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


Also defined as

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


Also see


Sources