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- Infimum_and_supremum abstract "In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set T is the greatest element that is less than or equal to all elements of S, if such an element exists. Consequently, the term greatest lower bound (abbreviated as GLB) is also commonly used.The supremum (abbreviated sup; plural suprema) of a subset S of a totally or partially ordered set T is the least element that is greater than or equal to all elements of S, if such an element exists. Consequently, the supremum is also referred to as the least upper bound (or LUB).The infimum is in a precise sense dual to the concept of a supremum. Infima and suprema of real numbers are common special cases that are important in analysis, and especially in Lebesgue integration. However, the general definitions remain valid in the more abstract setting of order theory where arbitrary partially ordered sets are considered.If the supremum of a subset S exists, it is unique. If S contains a greatest element, then that element is the supremum; otherwise, the supremum does not belong to S (or does not exist). Likewise, 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).The concepts of supremum and infimum are similar to minimum and maximum, but are more useful in analysis because they better characterize special sets which may have no minimum or maximum. For instance, the positive real numbers ℝ+ does not have a minimum, because any given element of ℝ+ could simply be divided in half resulting in a smaller number that is still in ℝ+. There is, however, exactly one infimum of the positive real numbers: 0, which is smaller than all the positive real numbers and greater than any other number which could be used as a lower bound. Note that 0 ∉ ℝ+.".
- Infimum_and_supremum thumbnail Infimum_illustration.svg?width=300.
- Infimum_and_supremum wikiPageID "39382".
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- Infimum_and_supremum wikiPageOutDegree "55".
- Infimum_and_supremum wikiPageRevisionID "683743461".
- Infimum_and_supremum wikiPageWikiLink Category:Order_theory.
- Infimum_and_supremum wikiPageWikiLink Complete_lattice.
- Infimum_and_supremum wikiPageWikiLink Complete_metric_space.
- Infimum_and_supremum wikiPageWikiLink Complete_space.
- Infimum_and_supremum wikiPageWikiLink Completeness_(order_theory).
- Infimum_and_supremum wikiPageWikiLink Completeness_of_the_real_numbers.
- Infimum_and_supremum wikiPageWikiLink Divisor.
- Infimum_and_supremum wikiPageWikiLink Duality_(order_theory).
- Infimum_and_supremum wikiPageWikiLink Empty_set.
- Infimum_and_supremum wikiPageWikiLink Essential_supremum_and_essential_infimum.
- Infimum_and_supremum wikiPageWikiLink Functional_(mathematics).
- Infimum_and_supremum wikiPageWikiLink Greatest_element.
- Infimum_and_supremum wikiPageWikiLink Hyperreal_number.
- Infimum_and_supremum wikiPageWikiLink Hyperreals.
- Infimum_and_supremum wikiPageWikiLink Integer.
- Infimum_and_supremum wikiPageWikiLink Irrational_number.
- Infimum_and_supremum wikiPageWikiLink Lattice.
- Infimum_and_supremum wikiPageWikiLink Lattice_(order).
- Infimum_and_supremum wikiPageWikiLink Least-upper-bound_property.
- Infimum_and_supremum wikiPageWikiLink Least_common_multiple.
- Infimum_and_supremum wikiPageWikiLink Lebesgue_integration.
- Infimum_and_supremum wikiPageWikiLink Limit_superior_and_limit_inferior.
- Infimum_and_supremum wikiPageWikiLink Lowest_common_multiple.
- Infimum_and_supremum wikiPageWikiLink Mathematical_analysis.
- Infimum_and_supremum wikiPageWikiLink Mathematics.
- Infimum_and_supremum wikiPageWikiLink Maxima_and_minima.
- Infimum_and_supremum wikiPageWikiLink Maximal_element.
- Infimum_and_supremum wikiPageWikiLink Maximum.
- Infimum_and_supremum wikiPageWikiLink Minimum.
- Infimum_and_supremum wikiPageWikiLink Natural_number.
- Infimum_and_supremum wikiPageWikiLink Order_theory.
- Infimum_and_supremum wikiPageWikiLink Partially_ordered_set.
- Infimum_and_supremum wikiPageWikiLink Positive_real_numbers.
- Infimum_and_supremum wikiPageWikiLink Power_set.
- Infimum_and_supremum wikiPageWikiLink Rational_number.
- Infimum_and_supremum wikiPageWikiLink Real_number.
- Infimum_and_supremum wikiPageWikiLink Real_numbers.
- Infimum_and_supremum wikiPageWikiLink Square_root_of_2.
- Infimum_and_supremum wikiPageWikiLink Subset.
- Infimum_and_supremum wikiPageWikiLink Total_order.
- Infimum_and_supremum wikiPageWikiLink Totally_ordered_set.
- Infimum_and_supremum wikiPageWikiLink Union_(set_theory).
- Infimum_and_supremum wikiPageWikiLink Well-order.
- Infimum_and_supremum wikiPageWikiLink File:Illustration_of_supremum.svg.
- Infimum_and_supremum wikiPageWikiLink File:Infimum_illustration.svg.
- Infimum_and_supremum wikiPageWikiLink File:Supremum_illustration.png.
- Infimum_and_supremum wikiPageWikiLinkText "Infimum and supremum".
- Infimum_and_supremum wikiPageWikiLinkText "greatest lower bound (infimum)".
- Infimum_and_supremum wikiPageWikiLinkText "infimum and supremum".
- Infimum_and_supremum wikiPageWikiLinkText "infimum function".
- Infimum_and_supremum wikiPageWikiLinkText "infimum".
- Infimum_and_supremum wikiPageWikiLinkText "supremum function".
- Infimum_and_supremum wikiPageWikiLinkText "supremum".
- Infimum_and_supremum hasPhotoCollection Infimum_and_supremum.
- Infimum_and_supremum id "p/u095810".
- Infimum_and_supremum title "Upper and lower bounds".
- Infimum_and_supremum wikiPageUsesTemplate Template:.
- Infimum_and_supremum wikiPageUsesTemplate Template:Commons_category.
- Infimum_and_supremum wikiPageUsesTemplate Template:Main.
- Infimum_and_supremum wikiPageUsesTemplate Template:Mathworld.
- Infimum_and_supremum wikiPageUsesTemplate Template:Overline.
- Infimum_and_supremum wikiPageUsesTemplate Template:Reflist.
- Infimum_and_supremum wikiPageUsesTemplate Template:Springer.
- Infimum_and_supremum subject Category:Order_theory.
- Infimum_and_supremum hypernym Element.
- Infimum_and_supremum type MilitaryUnit.
- Infimum_and_supremum comment "In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set T is the greatest element that is less than or equal to all elements of S, if such an element exists. Consequently, the term greatest lower bound (abbreviated as GLB) is also commonly used.The supremum (abbreviated sup; plural suprema) of a subset S of a totally or partially ordered set T is the least element that is greater than or equal to all elements of S, if such an element exists.".
- Infimum_and_supremum label "Infimum and supremum".
- Infimum_and_supremum sameAs Infimum_und_Supremum.
- Infimum_and_supremum sameAs Borne_supérieure_et_borne_inférieure.
- Infimum_and_supremum sameAs Estremo_superiore_e_estremo_inferiore.
- Infimum_and_supremum sameAs 상한과_하한.
- Infimum_and_supremum sameAs Kresy_dolny_i_górny.
- Infimum_and_supremum sameAs Supremo_e_ínfimo.
- Infimum_and_supremum sameAs m.09t31.
- Infimum_and_supremum sameAs Точная_верхняя_и_нижняя_границы_множеств.
- Infimum_and_supremum sameAs Инфимум_и_супремум.
- Infimum_and_supremum sameAs Q17502105.
- Infimum_and_supremum sameAs Q17502105.
- Infimum_and_supremum wasDerivedFrom Infimum_and_supremum?oldid=683743461.
- Infimum_and_supremum depiction Infimum_illustration.svg.
- Infimum_and_supremum isPrimaryTopicOf Infimum_and_supremum.