Definition:Upper Bound of Set

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 $\left({\R, \le}\right)$:

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

Also see

 * Definition:Bounded Above Set


 * Definition:Lower Bound of Set
 * Definition:Bounded Below Set


 * Definition:Bounded Ordered Set


 * Definition:Supremum of Set
 * Definition:Infimum of Set