Infimum and Supremum

note Consider moving this to analysis maybe

Definition

The infimum (abbr. , plural imfima) is a subset of a partially ordered set is the greatest element in that is less than or equal to each element of , if such an element exists. The term greatest lower bound or GLB is also used.

The supremum (abbr. , plural suprema) is a subset of a partially ordered set is the least element in that is greater than or equal to each element of , if such an element exists. The term lowest upper bound of LUB is also used.

Infimum and supremum are similar to minimum and maximum, but more useful because they work for sets which have no minimum or maximum.

For example, the set of positive real numbers does not have a minimum, because any number you chose can be divided by 2. But there exists exactly one infimum of the positive real numbers relative to the real numbers: 0. It is smaller than all positive real numbers, but the largest real number which could be used as a lower bound.

Important

An infimum must be defined relative to a superset of the set in question.

Formal Definition

A lower bound of a subset of a partially ordered set is an element such that for all .
A lower bound of is called an infimum if for all lower bounds of , ( is larger than all other lower bounds)

An upper bound of a subset of a partially ordered set is an element such that for all .
An upper bound of is called an supremum if for all upper bounds of , ( is smaller than all other upper bounds)