Upper and lower bounds


In mathematics, particularly in order theory, an upper bound or majorant of a subset of some partially ordered set is an element of which is greater than or equal to every element of.
Dually, a lower bound or minorant of is defined to be an element of which is less than or equal to every element of.
A set with an upper bound is said to be bounded from above or majorized by that bound.
The terms bounded above are also used in the mathematical literature for sets that have upper bounds.

Examples

For example, is a lower bound for the set , 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 numbers has a lower bound, since the natural numbers satisfy the well-ordering principle and thus have a least element. 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 numbers 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 partially ordered set as codomain, 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, function defined on domain and having the same codomain is an upper bound of, if for each in. 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 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, 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.