Bound

Pronunciation: /baʊnd/ ?

A lower bound of a set of numbers is a number such that the number is less than or equal all numbers in the set.[1] An upper bound of a set of numbers is a number such that the number is greater than or equal all numbers in the set.

The least upper bound of a set of numbers is the smallest number that is greater than or equal to all numbers in the set. The least upper bound is also called the supremum. A set is said to be bounded from above if it has a least upper bound.

The greatest lower bound of a set of numbers is the largest number that is less than or equal to all numbers in the set. The greatest lower bound is called the infimum. A set is said to be bounded from below if it has a greatest lower bound.

A set is bounded if it has an upper bound and a lower bound. A set is unbounded if it does not have an upper bound or does not have a lower bound.

Examples

SetGreatest Lower BoundLeast Upper BoundIs Bounded?
All real numbers {x: x∈ℝ}NoneNoneNo
All positive integers {x: x∈+ℤ}1NoneNo
{x: x∈ℝ, -1 < x < 5}-15Yes
{x: x∈ℝ, -1 ≤ x ≤ 5}-15Yes

Functions

When one says a function is bounded, one is referring to the range of the function. If the range of the function has a greatest lower bound, then one says that the function has a greatest lower bound.

References

  1. bounded. http://wordnet.princeton.edu/. WordNet. Princeton University. (Accessed: 2011-01-08). http://wordnetweb.princeton.edu/perl/webwn?s=bound&sub=Search+WordNet&o2=&o0=1&o7=&o5=&o1=1&o6=&o4=&o3=&h=.
  2. Veblen, Oswald and Lennes, N. J. . Introduction to Infintesimal Analysis; Functions of One Real Variable, 1st edition, pp 3-7. John Wiley and Sons, 1907. (Accessed: 2010-01-08). http://www.archive.org/stream/cu31924059316103#page/n18/mode/1up/search/bound.
  3. E. G. Phillips. Functions of a Complex Variable, Eight edition, pp 8-9. Oliver and Boyd, 1961. (Accessed: 2010-01-17). http://www.archive.org/stream/functionsofacomp029605mbp#page/n17/mode/1up/search/conjugate.

More Information

  • McAdams, David. Set. allmathwords.org. All Math Words Encyclopedia. Life is a Story Problem LLC. 2009-12-15. http://www.allmathwords.org/article.aspx?lang=en&id=Set.

Cite this article as:


Bound. 2010-01-04. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/b/bound.html.

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2010-10-02: Initial version (McAdams, David.)

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