Upper and lower bounds
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In mathematics, particularly in order theory, an upper bound or majorant[1] of a subset Template:Mvar of some preordered set (K, ≤)Script error: No such module "Check for unknown parameters". is an element of Template:Mvar that is greater than or equal to every element of Template:Mvar.[2][3] Dually, a lower bound or minorant of Template:Mvar is defined to be an element of Template:Mvar that is less than or equal to every element of Template:Mvar. 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]
Examples
For example, 5Script error: No such module "Check for unknown parameters". is a lower bound for the set S = Template:MsetScript error: No such module "Check for unknown parameters". (as a subset of the integers or of the real numbers, etc.), and so is 4Script error: No such module "Check for unknown parameters".. On the other hand, 6Script error: No such module "Check for unknown parameters". is not a lower bound for Template:Mvar since it is not smaller than every element in Template:Mvar. 13934Script error: No such module "Check for unknown parameters". and other numbers x such that x ≥ 13934Script error: No such module "Check for unknown parameters". would be an upper bound for S.
The set S = Template:MsetScript error: No such module "Check for unknown parameters". has 42Script error: No such module "Check for unknown parameters". as both an upper bound and a lower bound; all other numbers are either an upper bound or a lower bound for that Template:Mvar.
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.
Bounds of functions
The definitions can be generalized to functions and even to sets of functions.
Given a function Template:Italics correction with domain Template:Mvar and a preordered set (K, ≤)Script error: No such module "Check for unknown parameters". as codomain, an element Template:Mvar of Template:Mvar is an upper bound of Template:Italics correction if y ≥ Template:Italics correction(x)Script error: No such module "Check for unknown parameters". for each Template:Mvar in Template:Mvar. The upper bound is called sharp if equality holds for at least one value of Template:Mvar. It indicates that the constraint is optimal, and thus cannot be further reduced without invalidating the inequality.
Similarly, a function Template:Mvar defined on domain Template:Mvar and having the same codomain (K, ≤)Script error: No such module "Check for unknown parameters". is an upper bound of Template:Italics correction, if g(x) ≥ Template:Italics correction(x)Script error: No such module "Check for unknown parameters". for each Template:Mvar in Template:Mvar. The function Template:Mvar 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, 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.
Exact upper bounds
An upper bound Template:Mvar of a subset Template:Mvar of a preordered set (K, ≤)Script error: No such module "Check for unknown parameters". is said to be an exact upper bound for Template:Mvar if every element of Template:Mvar that is strictly majorized by Template:Mvar is also majorized by some element of Template:Mvar. Exact upper bounds of reduced products of linear orders play an important role in PCF theory.[5]