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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 Bound s (or ''suprema'') and Greatest Lower Bound s (or ''infima''). Another special kind of (least) upper bounds are Greatest Element s.

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 Cut s.

Further introductory information is found in the article on Order Theory .


BOUNDS OF FUNCTIONS


The definitions can be generalised to sets of Function s.

Let ''S'' be a set of functions S=\{f_1(\cdot), f_2(\cdot), \dots\}, with Domain ''F'' and having a partially ordered set as a Codomain .

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

In particular, g(\cdot) is said to be an ''upper bound'' of f(\cdot) when ''S'' consists of only one function f(\cdot) (i.e. ''S'' is a Singleton ). Note that this does not imply that f(\cdot) is a ''lower bound'' of g(\cdot).