In mathematics, especially in order theory, an upper bound of a subset S of some partially ordered set (P, ≤) is an element of P which is greater than or equal to every element of S. The term lower bound is defined dually as an element of P which is lesser 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 empty subset Φ of a partially ordered set P is conventionally considered to be both bounded from above and bounded from below with every element of P being both upper and lower bound of Φ. For example, if the number was 7.5, the lower bound would be 7.44999999 infinity but that cannot be expressed on a number line so it would have to be 7.45. The upper bound would be 7.55 and no higher.
Formally, given a partially ordered set (P, ≤), an element u of P is an upper bound of a subset S of P, if
Using ≥ instead of ≤ leads to the dual definition of a lower bound of S.
The bounds of a subset S of a partially ordered set P may or may not be elements of S itself. If S contains an upper bound then that upper bound is unique and is called the greatest element of S. The greatest element of S (if it exists) is also the least upper bound of S.
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.
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.
The definitions can be generalised to sets of functions.
A function with domain is an upper bound of S if for each function in the set and for each x in F.
In particular, is said to be an upper bound of when S consists of only one function (i.e. S is a singleton). Note that this does not imply that is a lower bound of .