Definition:Upper Bound of Set/Real Numbers
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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.
Also see
Sources
- 1964: Walter Rudin: Principles of Mathematical Analysis (2nd ed.) ... (previous) ... (next): Chapter $1$: The Real and Complex Number Systems: Real Numbers: $1.33$. Definition
- 1967: Michael Spivak: Calculus ... (previous): Part $\text {II}$: Foundations: Chapter $8$: Least Upper Bounds
- 1970: Arne Broman: Introduction to Partial Differential Equations ... (previous) ... (next): Chapter $1$: Fourier Series: $1.1$ Basic Concepts: $1.1.2$ Definitions
- 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $2$: Metric Spaces: $\S 5$: Limits: Definition $5.5$
- 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $1$: Review of some real analysis: $\S 1.1$: Real Numbers
- 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 2$: Continuum Property: $\S 2.2$: The Continuum Property
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 10$: The well-ordering principle
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): order properties (of real numbers)
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): bound