Definition:Upper Bound of Mapping/Real-Valued

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This page is about upper bounds of real-valued functions which are bounded above. For other uses, see Definition:Upper Bound.

Definition

Let $f: S \to \R$ be a real-valued function.


Let $f$ be bounded above in $\R$ by $H \in \R$.


Then $H$ is an upper bound of $f$.


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 $P \left({n}\right)$ is bounded above with the upper bound $N$

would be reported as:

The number $n$ such that $P \left({n}\right)$ 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.


Also see


Sources