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| {{Merge|Supremum|discuss=Talk:Infimum#Merge|date=July 2011}}
| | I'm Catherine and I live with my husband and our two children in Frettenham, in the south area. My hobbies are Vintage clothing, Vintage Books and Chainmail making.<br><br>Take a look at my blog: [http://www.platinum-djs.com/tag/dj-london/ Jason Hope christmas party] |
| [[Image:Infimum illustration.svg|thumb|right|300px|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 set]]s the infimum and the [[minimum]] are equal.]]
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| 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 number]]s are a common special case that is especially important in [[mathematical analysis|analysis]]. However, the general definition remains valid in the more abstract setting of [[order theory]] where arbitrary [[partially ordered set]]s are considered.
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| If the infimum exists, it is unique. If ''S'' contains a [[greatest element|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.
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| The infimum is in a precise sense [[duality (order theory)|dual]] to the concept of a [[supremum]].
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| == Infima of real numbers ==
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| In [[mathematical analysis|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 set|empty]], we define inf(''S'') = ∞ (see [[extended real number line]]).
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| 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).
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| Examples:
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| '''Simple'''
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| 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".
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| '''Complex'''
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| :<math>\inf\, \{1, 2, 3\} = 1.</math>
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| :<math>\inf\, \{ x \in \mathbb{R} : 0 < x < 1 \} = 0.</math>
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| :<math>\inf\, \{ x \in \mathbb{Q} : x^3 > 2 \} = \sqrt[3]{2}.</math>
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| :<math>\inf\, \{ (-1)^n + 1/n : n = 1, 2, 3, \dots \} = -1.</math> | |
| 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.
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| The notions of infimum and [[supremum]] are dual in the sense that
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| :<math>\inf(S) = -\sup(-S)</math>,
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| where
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| :<math>-S = \{ -s | s \in S \}.</math>
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| == Infima in partially ordered sets ==
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| The definition of infima easily generalizes to subsets of arbitrary [[partially ordered set]]s and as such plays a vital role in [[order theory]]. In this context, especially in [[lattice (order)|lattice theory]], greatest lower bounds are also called '''meets'''.
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| Formally, the ''infimum'' of a subset ''S'' of a partially ordered set (''P'', ≤) is an element ''a'' of ''P'' such that
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| # ''a'' ≤ ''x'' for all ''x'' in ''S'', (''a'' is a lower bound) and
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| # for all ''y'' in ''P'', if for all ''x'' in ''S'', ''y'' ≤ ''x'', then ''y'' ≤ ''a'' (''a'' larger than any other lower bound).
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| 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 (order theory)|completeness properties]].
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| The [[duality (order theory)|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 [[supremum|suprema]].
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| == Least upper bound property ==
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| See the article on the [[least-upper-bound property]].
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| == See also ==
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| * [[Essential suprema and infima]]
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| * [[Join and meet]]
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| * [[Limit superior and limit inferior]] (infimum limit)
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| * [[Supremum]]
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| == External links ==
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| * {{springer|title=Upper and lower bounds|id=p/u095810}}
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| * [http://planetmath.org/encyclopedia/Infimum.html Infimum] (''PlanetMath'')
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| * {{MathWorld |title=Infimum |id=Infimum}}
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| [[Category:Order theory]]
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I'm Catherine and I live with my husband and our two children in Frettenham, in the south area. My hobbies are Vintage clothing, Vintage Books and Chainmail making.
Take a look at my blog: Jason Hope christmas party