Upper and lower bounds

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A set with upper bounds and its least upper bound

In mathematics, particularly in order theory, an upper bound or majorant[1] of a subset S of some partially ordered set (K, ≤) is an element of K which 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 which 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]

Examples[edit]

For example, 5 is a lower bound for the set S = {5, 8, 42, 34, 13934} (as a subset of the integers or of the real numbers, etc.), and so is 4. On the other hand, 6 is not a lower bound for S since it is not smaller than every element in S.

The set S = {42} has 42 as both an upper bound and a lower bound; all other numbers are either an upper bound or a lower bound for that S.

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 (0, or 1 depending on the exact definition of natural numbers). 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[edit]

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

Given a function f with domain D and a partially ordered 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.[5]

Similarly, 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. 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 ≤.

Tight bounds[edit]

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.

References[edit]

  1. ^ a b Schaefer 1999, p. 3.
  2. ^ Mac Lane, Saunders; Birkhoff, Garrett (1991). Algebra. Providence, RI: American Mathematical Society. p. 145. ISBN 0-8218-1646-2.
  3. ^ "Upper Bound Definition (Illustrated Mathematics Dictionary)". www.mathsisfun.com. Retrieved 2019-12-03.
  4. ^ Weisstein, Eric W. "Upper Bound". mathworld.wolfram.com. Retrieved 2019-12-03.
  5. ^ "The Definitive Glossary of Higher Mathematical Jargon — Sharp". Math Vault. 2019-08-01. Retrieved 2019-12-03.