different between supremum vs residuum

English

Etymology

Borrowed from Latin supremum.

Noun

supremum (plural suprema)

1. (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.

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 ${\displaystyle S=\{t:t\in P:\forall x\in X,x\leq t\}}$ be the set of upper bounds of X. Then sup(X), if it exists, is the element ${\displaystyle s\in S:\forall y\in S,s\leq y}$.
  • The concept of supremum is closely related to the function ? (called join). The supremum of two elements, denoted ${\displaystyle \sup\{x,y\}}$ can also be written ${\displaystyle x\lor y}$. The supremum of a set may be denoted ${\displaystyle \sup(X)}$ or ${\displaystyle \bigvee X}$.

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

1. (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

1. (mathematics) supremum

Declension

Synonyms

• pienin yläraja

Antonyms

• infimum

Latin

Adjective

supr?mum

1. nominative neuter singular of supr?mus
2. accusative masculine singular of supr?mus
3. accusative neuter singular of supr?mus
4. 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

1. (mathematics) supremum

Declension

English

Etymology

From Latin residuum.

Pronunciation

• IPA^(key): /???z?.dju.?m/

Noun

residuum (plural residuums or residua)

1. The residue, remainder or rest of something
2. (chemistry) The solid material remaining after the liquid in which it was dissolved has been evaporated; a residue.
3. (fuzzy logic) A binary function from [0,1] × [0,1] to [0,1] which is defined in terms of the t-norm as follows: ${\displaystyle x\rightarrow y=\sup\{z|z*x\leqslant y\}}$, where ${\displaystyle *}$ denotes the t-norm function and ${\displaystyle \sup }$ denotes the supremum.

   The residuum of the minimum t-norm is a function ${\displaystyle \rightarrow }$ such that ${\displaystyle x\rightarrow y={\begin{cases}y,&{\text{if }}x>y\\1,&{\text{if }}x\leqslant y.\end{cases}}}$

4. (law) The residue of an estate

Synonyms

• (rest of something): lave, remnant; See also Thesaurus:remainder

See also

• (fuzzy logic): relative pseudo-complement

Latin

Adjective

residuum

1. nominative neuter singular of residuus
2. accusative masculine singular of residuus
3. accusative neuter singular of residuus
4. vocative neuter singular of residuus

References

• residuum in Charles du Fresne du Cange’s Glossarium Mediæ et Infimæ Latinitatis (augmented edition, 1883–1887)