Definition:Upper Bound of Mapping/Real-Valued
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This page is about Upper Bound of Real-Valued Function. For other uses, see 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 $\map P n$ 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.
Also see
Sources
- 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 7.13$
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): bound: 1. (of a function)
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): bound: 1. (of a function)