different between supremum vs infimum
supremum
English
Etymology
Borrowed from Latin supremum.
Noun
supremum (plural suprema)
- (set theory) (real analysis): Given a subset X of R, the smallest real number that is ? every element of X; (order theory): given a subset X of a partially ordered set P (with partial order ?), the least element y of P such that every element of X is ? y.
- 2006, Charalambos D. Aliprantis, Kim C. Border, Infinite Dimensional Analysis: A Hitchhiker's Guide, Springer, 3rd Edition, page 8,
- A sublattice of a lattice is a subset that is closed under pairwise infima and suprema.
- 2010, James S. Howland, Basic Real Analysis, Jones & Bartlett Publishers, page 9,
- The best way to describe the supremum of S is to say that it wants to be the greatest element of S. In fact, if S has a greatest element, then that element is the supremum.
- 2011, Andreas Löhne, Vector Optimization with Infimum and Supremum, Springer, page vii,
- The key to an approach to vector optimization based on infimum and supremum is to consider set-based objective functions and to extend the partial ordering of the original objective space to a suitable subspace of the power set. In this new space the infimum and supremum exist under the usual assumptions.
- 2006, Charalambos D. Aliprantis, Kim C. Border, Infinite Dimensional Analysis: A Hitchhiker's Guide, Springer, 3rd Edition, page 8,
Usage notes
- Commonly denoted sup(X).
- The supremum of X may not exist, and, if it does, may not be an element of X.
- (order theory):
- Formally: Let be the set of upper bounds of X. Then sup(X), if it exists, is the element .
- The concept of supremum is closely related to the function ? (called join). The supremum of two elements, denoted can also be written . The supremum of a set may be denoted or .
Synonyms
- (element of a set greater than or equal to all members of a given subset): least upper bound, LUB, sup
Coordinate terms
- infimum
Translations
See also
- maximum
Further reading
- Infimum and supremum on Wikipedia.Wikipedia
- Join and meet on Wikipedia.Wikipedia
- Lattice (order) on Wikipedia.Wikipedia
- Least-upper-bound property on Wikipedia.Wikipedia
- Upper and lower bounds on Wikipedia.Wikipedia
Anagrams
- supermum
Czech
Pronunciation
- IPA(key): [?supr?mum]
Noun
supremum n
- (mathematics) supremum
Antonyms
- infimum
Further reading
- supremum in Akademický slovník cizích slov, 1995, at prirucka.ujc.cas.cz
Finnish
Pronunciation
- IPA(key): /?supre(?)mum/, [?s?upre?(?)mum]
- Rhymes: -upremum
- Syllabification: sup?re?mum
Noun
supremum
- (mathematics) supremum
Declension
Synonyms
- pienin yläraja
Antonyms
- infimum
Latin
Adjective
supr?mum
- nominative neuter singular of supr?mus
- accusative masculine singular of supr?mus
- accusative neuter singular of supr?mus
- vocative neuter singular of supr?mus
References
- supremum in Charlton T. Lewis (1891) An Elementary Latin Dictionary, New York: Harper & Brothers
- supremum in Gaffiot, Félix (1934) Dictionnaire illustré Latin-Français, Hachette
- Carl Meissner; Henry William Auden (1894) Latin Phrase-Book?[1], London: Macmillan and Co.
Swedish
Noun
supremum n
- (mathematics) supremum
Declension
supremum From the web:
- what supremum means
- what is supremum and infimum of r is
- what is supremum and infimum of a set
- what is supremum distance
- what is supremum norm
- what is supremum and infimum in mathematics
- what does supremum norm mean
- what does supremum and infimum mean
infimum
English
Etymology
Borrowed from Latin infimum (“lowest part”), from infimus (“lowest”).
Pronunciation
- (UK) IPA(key): /?n?fa?m?m/
- (US) IPA(key): /?n?fi?m?m/, /?n?fa?m?m/
Noun
infimum (plural infima or infimums)
- (mathematics) (of a subset) the greatest element of the containing set that is smaller than or equal to all elements of the subset. The infimum may or may not be a member of the subset.
Synonyms
- greatest lower bound
Antonyms
- supremum
Translations
Czech
Pronunciation
- IPA(key): [??nf?mum]
Noun
infimum n
- (mathematics) infimum
Antonyms
- supremum
Further reading
- infimum in Akademický slovník cizích slov, 1995, at prirucka.ujc.cas.cz
Finnish
Pronunciation
- IPA(key): /?infi(?)mum/, [?infi(?)mum]
- Rhymes: -infimum
- Syllabification: in?fi?mum
Noun
infimum
- (mathematics) infimum
Declension
Synonyms
- suurin alaraja
Antonyms
- supremum
Latin
Adjective
?nfimum
- nominative neuter singular of ?nfimus
- accusative masculine singular of ?nfimus
- accusative neuter singular of ?nfimus
- vocative neuter singular of ?nfimus
infimum From the web:
- what infimum means
- what is infimum and supremum in real analysis
- what is infimum of a function
- what does infimum
- what is the infimum of 1/n
- what is limit infimum
- what is the infimum of sequence
- what does inf infimum mean
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