mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, 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, a 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 a ...

of some preordered set is an element of that is 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 (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...

or of the

real numbers In mathematics, a real number is a number that can be used to measurement, measure a continuous variable, continuous one-dimensional quantity such as a time, duration or temperature. Here, ''continuous'' means that pairs of values can have arbi ...

, etc.), and so is . On the other hand, is not a lower bound for since it is not smaller than every element in . and other numbers ''x'' such that would be an upper bound for ''S''. 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 the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...

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 integers 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 (for example, The set of all ...

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 order 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 ( ref ...

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, a codomain, counter-domain, or set of destination of a function is a 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 ...

, 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; : infima) of a subset S of a partially ordered set P is the greatest element in P that is less than or equal to each element of S, if such an element exists. If the infimum of S exists, it is unique, ...

'', 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; : infima) of a subset S of a partially ordered set P is the greatest element in P that is less than or equal to each element of S, if such an element exists. If the infimum of S exists, it is unique ...

'', 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 , last1=Schaefer , first1=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

Examples

Bounds of functions

Tight bounds

Exact upper bounds

See also

References