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. Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations related to: Mathematics Look up Mathematics on Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ...
Order theory is a branch of mathematics that studies various kinds of binary relations that capture the intuitive notion of a mathematical ordering. ...
In mathematics, especially order theory, a partially ordered set (or poset for short) is a set equipped with a partial order relation. ...
In the mathematical area of order theory, every partially ordered set P gives rise to a dual (or opposite) partially ordered set which is often denoted by Pop. ...
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. Natural number can mean either a positive integer (1, 2, 3, 4, ...) or a nonnegative integer (0, 1, 2, 3, 4, ...). Natural numbers have two main purposes: they can be used for counting (there are 3 apples on the table), or they can be used for ordering (this is...
In mathematics, the supremum of an ordered set S is the least element (not necessarily in S) which is greater than or equal to each element of S. Consequently, it is also referred to as the least upper bound. ...
In mathematics the infimum of a subset of some set is the greatest element, not necessarily in the subset, that is smaller than all other elements of the subset. ...
In mathematics, especially in order theory, the greatest element of a subset S of a partially ordered set is an element of S which is greater than or equal to any other element of S. The term least element is defined dually. ...
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. In mathematics, a Dedekind cut, named after Richard Dedekind, in a totally ordered set S is a partition of it, (A, B), such that A is closed downwards (meaning that for all a in A, x â‰¤ a implies that x is in A as well) and B is closed upwards...
Further introductory information is found in the article on order theory. Order theory is a branch of mathematics that studies various kinds of binary relations that capture the intuitive notion of a mathematical ordering. ...
Bounds of functions
The definitions can be generalised to sets of functions. In mathematics, a function is a relation, such that each element of a set (the domain) is associated with a unique element of another (possibly the same) set (the codomain, not to be confused with the range). ...
Let S be a set of functions , with domain F and having a partially ordered set as a codomain. In mathematics, the domain of a function is the set of all input values to the function. ...
A codomain in mathematics is the set of output values associated with (or mapped to) the domain of inputs in a function. ...
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 . In mathematics, a singleton is a set with exactly one element. ...
