Upper and lower bounds

In mathematics, particularly in order theory, an upper bound or majorant[1] of a subset S of some preordered set (K, ≤) is an element of K that is greater than or equal to every element of S.[2][3] Dually, a lower bound or minorant of S is defined to be an element of K that is less than or equal to every element of S. A set with an upper (respectively, lower) bound is said to be bounded from above or majorized[1] (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.[4]

Every subset of the natural 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 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.

The definitions can be generalized to functions and even to sets of functions.

Given a function f with domain D and a preordered set (K, ≤) as codomain, an element y of K is an upper bound of f if yf(x) for each x in D. The upper bound is called sharp if equality holds for at least one value of x. It indicates that the constraint is optimal, and thus cannot be further reduced without invalidating the inequality.

Similarly, a function g defined on domain D and having the same codomain (K, ≤) is an upper bound of f, if g(x) ≥ f(x) for each x in D. The function g 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 ≤.

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.

An upper bound u of a subset S of a preordered set (K, ≤) is said to be an exact upper bound for S if every element of K that is strictly majorized by u is also majorized by some element of S. Exact upper bounds of reduced products of linear orders play an important role in PCF theory.[5]