Definition:Upper Bound of Set

From ProofWiki
Jump to navigation Jump to search

This page is about upper bounds of ordered sets which are bounded above. For other uses, see Definition:Upper Bound.

Definition

Let $\left({S, \preceq}\right)$ 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 $\left({S, \preceq}\right)$ 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 $\R$.


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.


Also see


Sources