Definition:Upper Bound of Set/Real Numbers

This page is about Upper Bound of Subset of Real Numbers. For other uses, see Upper Bound.

Definition

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 $T$.

Upper Bound of Number

When considering the upper bound of a set of numbers, it is commonplace to ignore the set and instead refer just to the number itself.

Thus the construction:

The set of numbers which fulfil the propositional function $\map P n$ is bounded above with the upper bound $N$

would be reported as:

The number $n$ such that $\map P n$ has the upper bound $N$.

This construct obscures the details of what is actually being stated. Its use on $\mathsf{Pr} \infty \mathsf{fWiki}$ is considered an abuse of notation and so discouraged.

This also applies in the case where it is the upper bound of a mapping which is under discussion.