supremum

== English == === Etymology === Borrowed from Latin supremum. === Pronunciation === (Received Pronunciation) IPA(key): /səˈpɹiːˌmʌm/ (General American, Canada) IPA(key): /səˈpɹiˌmʌm/ (General Australian, New Zealand) IPA(key): /səˈpɹiːˌmɐm/ === Noun === supremum (plural supremums or 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. Synonyms: least upper bound, LUB, sup Coordinate term: infimum ==== 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 S = { t : t ∈ P : ∀ x ∈ X , x ≤ t } {\displaystyle S=\{t:t\in P:\forall x\in X,x\leq t\}} be the set of upper bounds of X. Then sup(X), if it exists, is the element s ∈ S : ∀ y ∈ S , s ≤ y {\displaystyle s\in S:\forall y\in S,s\leq y} . The concept of supremum is closely related to the function ∨ (called join). The supremum of two elements, denoted sup { x , y } {\displaystyle \sup\{x,y\}} can also be written x ∨ y {\displaystyle x\lor y} . The supremum of a set may be denoted sup ( X ) {\displaystyle \sup(X)} or ⋁ X {\displaystyle \bigvee X} . ==== Derived terms ==== ==== Translations ==== === See also === === 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): [ˈsuprɛːmum] === Noun === supremum n (mathematics) supremum Antonym: infimum ==== Declension ==== === Further reading === “supremum”, in Akademický slovník cizích slov at prirucka.ujc.cas.cz [Academic dictionary of foreign words] (in Czech), 1995 == Finnish == === Etymology === Learned borrowing from Latin suprēmum. === Pronunciation === IPA(key): /ˈsupre(ː)mum/, [ˈs̠upre̞(ː)mum] Rhymes: -upremum Syllabification(key): sup‧re‧mum Hyphenation(key): sup‧re‧mum === Noun === supremum (mathematics) supremum ==== Declension ==== ==== Synonyms ==== pienin yläraja ==== Antonyms ==== infimum == Latin == === Adjective === suprēmum inflection of suprēmus: nominative/accusative/vocative neuter singular accusative masculine singular === 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 ====