Upper bound

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In mathematics, especially in order theory, an upper bound of a subset S of some partially ordered set is an element which is greater than or equal to every element of S. The term lower bound is defined dually.

Formally, given a partially ordered set (P, ≤), an element u of P is an upper bound of a subset S of P, if

s ≤ u, for all elements s of S.

Using ≥ instead of ≤ leads to the dual definition of a lower bound of S.

Clearly, a subset of a partially ordered set may fail to have any upper bounds. Consider for example the subset of the natural numbers which are greater than a given natural number. On the other hand, a set may have many different upper and lower bounds, and hence one is usually interested in picking out specific elements from the sets of upper or lower bounds. This leads to the consideration of least upper bounds (or suprema) and greatest lower bounds (or infima). Another special kind of (least) upper bounds are greatest elements.

A special situation does occur when a subset is equal to the set of lower bounds of its own set of upper bounds. This observation leads to the definition of Dedekind cuts.

Further introductory information is found in the article on order theory.

Bounds of functions

The definitions can be generalised to sets of functions.

Let S be a set of functions <math>S=\{f_1(\cdot), f_2(\cdot), \dots\}</math>, with domain F and having a partially ordered set as a codomain.

A function <math>g(\cdot)</math> with domain <math>G \supseteq F</math> is an upper bound of S if <math>f_i(x) \le g(x)</math> for each function <math>f_i(\cdot)</math> in the set and for each x in F.

In particular, <math>g(\cdot)</math> is said to be an upper bound of <math>f(\cdot)</math> when S consists of only one function <math>f(\cdot)</math> (i.e. S is a singleton). Note that this does not imply that <math>f(\cdot)</math> is a lower bound of <math>g(\cdot)</math>.es:Mayorante fr:Majorant he:חסם (מתמטיקה) it:Maggiorante zh:上界