Upper and Lower Bound

References

(Wikipedia, 2015) ⇒ http://wikipedia.org/wiki/Upper_and_lower_bounds
- QUOTE: In mathematics, especially 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 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.

Bounds of functions, 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, \leq)

as codomain, an element

y

of

K

is a upper bound of

f

if

y \geq f (x)

for each

x

in

D

. The upper bound is called sharp if equality holds for at least one value of

x

.

Function

g

defined on domain

D

and having the same codomain

(K, \leq)

is an upper bound of

f

if

g (x) \geq f (x)

for each

x

in

D

.

Function

g

is further said to be a 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, with ≤ replacing ≥.

(Wolfram Mathworld , 1999) ⇒ http://mathworld.wolfram.com/UpperBound.html
- QUOTE: A function [math]\displaystyle{ f }[/math] is said to have a upper bound [math]\displaystyle{ C }[/math] if [math]\displaystyle{ f(x)\leq C }[/math]for all [math]\displaystyle{ x }[/math] in its domain. The least upper bound is called the supremum. A set is said to be bounded from above if it has an upper bound.
(Wolfram Mathworld , 1999) ⇒ http://mathworld.wolfram.com/LowerBound.html
- QUOTE: A function [math]\displaystyle{ f }[/math] is said to have a lower bound [math]\displaystyle{ c }[/math] if [math]\displaystyle{ c \leq f(x) }[/math] for all <math> x <math> in its domain. The greatest lower bound is called the infimum.