Upper and lower bounds

  (Redirected from Lower bound)
A set with upper bounds and its least upper bound.

In mathematics, particularly in order theory, an upper bound 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.[1][2] In a similar manner, the term lower bound is defined dually as an element of K which is less than or equal to every element of S. A set with an upper bound is said to be bounded from above by that bound, a set with a lower bound is said to be bounded from below by that bound. The terms bounded above (bounded below) are also used in the mathematical literature for sets that have upper (respectively lower) bounds.[3]

ExamplesEdit

For example, 5 is a lower bound for the set S = { 5, 8, 42, 34, 13934 }; so would 4 be if 4 belongs to the set K. On the other hand, 6 is not a lower bound for S, as it is not smaller than every element in S.

As another example, consider the set S = { 42 }. Here, the number 42 is both an upper bound and a lower bound of S, and all other real numbers are either an upper bound or a lower bound for that set.

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 functionsEdit

The definitions can be generalized to functions and even 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.[4]

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 boundsEdit

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.

ReferencesEdit

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