Infimum and supremum

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A set T of real numbers (red and green balls), a subset S of T (green balls), and the infimum of S. Note that for finite sets the infimum and the minimum are equal.

In mathematics, the infimum (plural infima) of a subset S of a partially ordered set T is the greatest element of T that is less than or equal to all elements of S. Consequently the term greatest lower bound (abbreviated as glb or GLB) 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.

If the infimum exists, it is unique. If S contains a least element, then that element is the infimum; otherwise, the infimum does not belong to S (or does not exist). For instance, the positive real numbers do not have a least element, and their infimum is 0, which is not a positive real number.

The infimum is in a precise sense dual to the concept of a supremum.

Infima of real numbers[edit]

In analysis the infimum or greatest lower bound of a subset 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:

Simple

The "Infimum" or "Greatest Lower Bound" of the set of numbers 2, 3, 4 is 2. 1 would be a lower bound but not the "greatest lower bound" and hence not the "Infimum".

Complex

\inf\, \{1, 2, 3\} = 1.
\inf\, \{ x \in \mathbb{R} : 0 < x < 1 \}  =  0.
\inf\, \{ x \in \mathbb{Q} : x^3 > 2 \} = \sqrt[3]{2}.
\inf\, \{ (-1)^n + 1/n : n = 1, 2, 3, \dots \} = -1.

If a set has a smallest element, as in the first example, then the smallest element is the infimum for the set. (If the infimum is contained in the set, then it is also known as the minimum). As the last three examples show, the infimum of a set does not have to belong to the set.

The notions of infimum and supremum are dual in the sense that

\inf(S) = -\sup(-S),

where

-S = \{ -s | s \in S \}.

Infima in partially ordered sets[edit]

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 a of P such that

  1. ax for all x in S, (a is a lower bound) and
  2. for all y in P, if for all x in S, yx, then ya (a larger than any other lower bound).

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.

Least upper bound property[edit]

See the article on the least-upper-bound property.

See also[edit]

External links[edit]