Inmathematics, particularly inorder theory, anupper bound ormajorant[1] of asubsetS of somepreordered set(K, ≤) is anelement ofK that isgreater than or equal to every element ofS.[2][3]Dually, alower bound orminorant ofS is defined to be an element ofK that is less than or equal to every element ofS. A set with an upper (respectively, lower) bound is said to bebounded from above ormajorized[1] (respectivelybounded from below orminorized) by that bound. The termsbounded above (bounded below) are also used in the mathematical literature for sets that have upper (respectively lower) bounds.[4]
For example,5 is a lower bound for the setS = {5, 8, 42, 34, 13934} (as a subset of theintegers or of thereal numbers, etc.), and so is4. On the other hand,6 is not a lower bound forS since it is not smaller than every element inS.13934 and other numbersx such thatx ≥ 13934 would be an upper bound forS.
The setS = {42} has42 as both an upper bound and a lower bound; all other numbers are either an upper bound or a lower bound for thatS.
Every subset of thenatural numbers 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 therational numbers may or may not be bounded from below, and may or may not be bounded from above.
Every finite subset of a non-emptytotally ordered set has both upper and lower bounds.
The definitions can be generalized tofunctions and even to sets of functions.
Given a functionf withdomainD and a preordered set(K, ≤) ascodomain, an elementy ofK is an upper bound off ify ≥f(x) for eachx inD. The upper bound is calledsharp if equality holds for at least one value ofx. It indicates that the constraint is optimal, and thus cannot be further reduced without invalidating the inequality.
Similarly, a functiong defined on domainD and having the same codomain(K, ≤) is an upper bound off, ifg(x) ≥f(x) for eachx inD. The functiong is further said to be an upper bound of a set of functions, if it is an upper bound ofeach function in that set.
The notion of lower bound for (sets of) functions is defined analogously, by replacing ≥ with ≤.
An upper bound is said to be atight upper bound, aleast upper bound, or asupremum, if no smaller value is an upper bound. Similarly, a lower bound is said to be atight lower bound, agreatest lower bound, or aninfimum, if no greater value is a lower bound.
An upper boundu of a subsetS of a preordered set(K, ≤) is said to be anexact upper bound forS if every element ofK that is strictly majorized byu is also majorized by some element ofS. Exact upper bounds ofreduced products oflinear orders play an important role inPCF theory.[5]