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 intr ...

, 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 In mathematics, especially in order theory, a preorder or quasiorder is a binary relation that is reflexive relation, reflexive and Transitive relation, transitive. Preorders are more general than equivalence relations and (non-strict) partia ...

is an element of that is

greater than or equal to In mathematics, an inequality is a relation which makes a non-equal comparison between two numbers or other mathematical expressions. It is used most often to compare two numbers on the number line by their size. There are several different ...

every element of .

Dually Dually may refer to: *Dualla, County Tipperary, a village in Ireland *A pickup truck with dual wheels on the rear axle * DUALLy, s platform for architectural languages interoperability * Dual-processor See also * Dual (disambiguation) Dual or ...

, 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 o ...

or of the

real numbers In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...

, 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 ...

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 of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ration ...

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 In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X: # a \leq a ( reflexive) ...

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 Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined **Domain of definition of a partial function **Natural domain of a partial function **Domain of holomorphy of a function * Do ...

and a preordered set as

codomain In mathematics, the codomain or set of destination of a function is the set into which all of the output of the function is constrained to fall. It is the set in the notation . The term range is sometimes ambiguously used to refer to either the ...

, an element of is an upper bound of if for each in . The upper bound is called ''

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'' 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 l ...

'', 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 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 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.

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 mathematical research and scholarship, and serves the national and international community through its publications, meetings, ...

, 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