# Definition:Upper Bound of Set

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

## Definition

Let $\struct {S, \preceq}$ be an ordered set.

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

An upper bound for $T$ (in $S$) is an element $M \in S$ such that:

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

That is, $M$ succeeds 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 $\R$.

An upper bound for $T$ (in $\R$) is an element $M \in \R$ such that:

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

That is, $M$ is greater than or equal to every element of $T$.

## Also defined as

Some sources use the terminology the upper bound to refer to the notion of supremum.