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In mathematics, particularly in
order theory Order theory is a branch of mathematics that investigates the intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article intro ...
, an upper bound or majorant of a
subset In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...
of some preordered set is an element of that is greater than or equal to every element of . Dually, a lower bound or minorant of is defined to be an element of that is less than or equal to every element of . A set with an upper (respectively, lower) bound is said to be bounded from above or majorized (respectively bounded from below or minorized) by that bound. The terms bounded above (bounded below) are also used in the mathematical literature for sets that have upper (respectively lower) bounds.


Examples

For example, is a lower bound for the set (as a subset of the
integers An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language of ...
or of the
real numbers In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
, etc.), and so is . On the other hand, is not a lower bound for since it is not smaller than every element in . The set has as both an upper bound and a lower bound; all other numbers are either an upper bound or a lower bound for that . Every subset of the
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''Cardinal n ...
s has a lower bound since the natural numbers have a least element (0 or 1, depending on convention). An infinite subset of the natural numbers cannot be bounded from above. An infinite subset of the
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language of ...
s may be bounded from below or bounded from above, but not both. An infinite subset of the
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction (mathematics), fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ) ...
s may or may not be bounded from below, and may or may not be bounded from above. Every finite subset of a non-empty totally ordered set has both upper and lower bounds.


Bounds of functions

The definitions can be generalized to functions and even to sets of functions. Given a function with domain and a preordered set as
codomain In mathematics, the codomain or set of destination of a Function (mathematics), function is the Set (mathematics), set into which all of the output of the function is constrained to fall. It is the set in the notation . The term Range of a funct ...
, an element of is an upper bound of if for each in . The upper bound is called '' sharp'' if equality holds for at least one value of . It indicates that the constraint is optimal, and thus cannot be further reduced without invalidating the inequality. Similarly, a function defined on domain and having the same codomain is an upper bound of , if for each in . The function is further said to be an upper bound of a set of functions, if it is an upper bound of ''each'' function in that set. The notion of lower bound for (sets of) functions is defined analogously, by replacing ≥ with ≤.


Tight bounds

An upper bound is said to be a ''tight upper bound'', a ''least upper bound'', or a ''
supremum In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest low ...
'', if no smaller value is an upper bound. Similarly, a lower bound is said to be a ''tight lower bound'', a ''greatest lower bound'', or an '' infimum'', if no greater value is a lower bound.


Exact upper bounds

An upper bound of a subset of a preordered set is said to be an ''exact upper bound'' for if every element of that is strictly majorized by is also majorized by some element of . Exact upper bounds of reduced products of linear orders play an important role in PCF theory.


See also

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Greatest element and least element In mathematics, especially in order theory, the greatest element of a subset S of a partially ordered set (poset) is an element of S that is greater than every other element of S. The term least element is defined duality (order theory), dually, ...
*
Infimum and supremum In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest low ...
*
Maximal and minimal elements In mathematics, especially in order theory, a maximal element of a subset ''S'' of some preordered set is an element of ''S'' that is not smaller than any other element in ''S''. A minimal element of a subset ''S'' of some preordered set is define ...


References

{{reflist , refs= {{cite book , last1 = Mac Lane, first1 = Saunders , author1-link = Saunders Mac Lane , last2 = Birkhoff, first2 = Garrett , author2-link = Garrett Birkhoff , title = Algebra , url = https://archive.org/details/algebra00lane, url-access = limited, place = Providence, RI , publisher =
American Mathematical Society The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematics, mathematical research and scholarship, and serves the national and international community through its publicatio ...
, page
145
, year = 1991 , isbn = 0-8218-1646-2
{{cite book , last=Schaefer , first=Helmut H. , author-link=Helmut H. Schaefer , last2=Wolff , first2=Manfred P. , title=Topological Vector Spaces , publisher=Springer New York Imprint Springer , series= GTM , volume=8 , page=3 , publication-place=New York, NY , year=1999 , isbn=978-1-4612-7155-0 , oclc=840278135 Mathematical terminology Order theory Real analysis de:Schranke (Mathematik) pl:Kresy dolny i górny