Upper and lower bounds
In mathematics, particularly in
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]Examples
For example, 5 is a lower bound for the set S = {5, 8, 42, 34, 13934} (as a subset of the
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 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
Bounds of functions
The definitions can be generalized to functions and even to sets of functions.
Given a function f with
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 ≤.
Tight bounds
An upper bound is said to be a tight upper bound, a least upper bound, or a
Exact upper bounds
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
See also
References
- ^ OCLC 840278135.
- ISBN 0-8218-1646-2.
- ^ "Upper Bound Definition (Illustrated Mathematics Dictionary)". Math is Fun. Retrieved 2019-12-03.
- ^ Weisstein, Eric W. "Upper Bound". mathworld.wolfram.com. Retrieved 2019-12-03.
- ^ Kojman, Menachem. "Exact upper bounds and their uses in set theory".