Infimum
|
|
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. Consequently the term greatest lower bound is also commonly used. Infima of real numbers are a common special case that is especially important in analysis. However, the general definition remains valid in the more abstract setting of order theory where arbitrary partially ordered sets are considered.
Infima are in a precise sense dual to the concept of a supremum and thus additional information and examples are found within the corresponding article.
Infima of real numbers
In analysis the infimum or greatest lower bound of a set S of real numbers is denoted by inf(S) and is defined to be the biggest real number that is smaller than or equal to every number in S. If no such number exists (because S is not bounded below), then we define inf(S) = -∞. If S is empty, we define inf(S) = ∞ (see extended real number line).
An important property of the real numbers is that every set of real numbers has an infimum (any bounded nonempty subset of the real numbers has an infimum in the non-extended real numbers).
Examples:
- <math>\inf \{ x \in \mathbb{R} | 0 < x < 1 \} = 0<math>
- <math>\inf \{ x \in \mathbb{R} | x^3 > 2 \} = 2^{1/3}<math>
- <math>\inf \{ (-1)^n + 1/n | n = 1, 2, 3, \dots \} = -1<math>
Note that the infimum does not have to belong to the set (like in these examples). If the infimum value belongs to the set then we can say there is a smallest element in the set.
The infimum and supremum of S are related via
- <math>\inf(S) = -\sup(-S)<math>.
In general, in order to show that inf(S) ≥ A, one only has to show that x ≥ A for all x in S. Showing that inf(S) ≤ A is a bit harder: for any ε > 0, you have to exhibit an element x in S with x ≤ A + ε (of course, if you can find an element x in S with x ≤ A, you are done right away).
See also: limit inferior.
Infima within partially ordered sets
The definition of infima easily generalizes to subsets of arbitrary partially ordered sets and as such plays a vital role in order theory. In this context, especially in lattice theory, greatest lower bounds are also called meets.
Formally, the infimum of a subset S of a partially ordered set (P, ≤) is an element l of P such that
- l ≤ x for all x in S, and
- for any p in P such that p ≤ x for all x in S it holds that p ≤ l.
Any element with these properties is necessarily unique, but in general no such element needs to exist. Consequently, orders for which certain infima are known to exist become especially interesting. More information on the various classes of partially ordered sets that arise from such considerations are found in the article on completeness properties.
The dual concept of infimum is given by the notion of a supremum or least upper bound. By the duality principle of order theory, every statement about suprema is thus readily transformed into a statement about infima. For this reason, all further results, details, and examples can be taken from the article on suprema.
Greatest lower bound property
See the article on the least upper bound property.
For information on the Intermediate-Range Nuclear Forces Treaty, go here: INF