different between relation vs antisymmetric

English

Etymology

From Middle English relacion, relacioun, from Anglo-Norman relacioun and Old French relacion (whence French relation), from Latin rel?ti?, noun of process form from perfect passive participle rel?tus (“related”), from verb refer? (“I refer, I relate”), from prefix re- (“again”) + fer? (“I bear, I carry”).

Morphologically relate +? -ion

Pronunciation

• enPR: r?-l?'sh?n, IPA^(key): /???le???n/
• Rhymes: -e???n

Noun

relation (countable and uncountable, plural relations)

1. The manner in which two things may be associated.

   • Carried somehow, somewhither, for some reason, on these surging floods, were these travelers, of errand not wholly obvious to their fellows, yet of such sort as to call into query alike the nature of their errand and their own relations. It is easily earned repetition to state that Josephine St. Auban's was a presence not to be concealed.

2. A member of one's extended family; a relative.

3. The act of relating a story.
   • 1669, Letter from Dr. Merrett to Thomas Browne, in Simon Wilkin (ed.), Sir Thomas Browne’s Works including his Life and Correspondence, London: William Pickering, 1836, Volume I, p. 443,^[1]

     Many of the lupus piscis I have seen, and have bin informed by the king’s fishmonger they are taken on our coast, but was not satisfied for some reasons of his relation soe as to enter it into my Pinax […]

   • 1691, Arthur Gorges (translator), The Wisdom of the Ancients by Francis Bacon (1609), London, Preface,^[2]

     […] seeing they are diversly related by Writers that lived near about one and the self-same time, we may easily perceive that they were common things, derived from precedent Memorials; and that they became various, by reason of the divers Ornaments bestowed on them by particular Relations […]
4. (set theory) A set of ordered tuples.

   • […] Signs are, first of all, physical things: for example, chalk marks on a blackboard, pencil or ink marks on paper, sound waves produced in a human throat. According to Reichenbach, "What makes them signs is the intermediary position they occupy between an object and a sign user, i.e., a person." For a sign to be a sign, or to function as such, it is necessary that the person take account of the object it designates. Thus, anything in nature may or may not be a sign, depending on a person's attitude toward it. A physical thing is a sign when it appears as a substitute for, or representation of, the object for which it stands with respect to the sign user. The three-place relation between sign, object, and sign user is called the sign relation or relation of denotation.

5. (set theory) Specifically, a set of ordered pairs; a binary relation.

6. (databases) A set of ordered tuples retrievable by a relational database; a table.

7. (mathematics) A statement of equality of two products of generators, used in the presentation of a group.
8. (category theory) A subobject of a product of objects.
9. (usually collocated: sexual relation) The act of intercourse.

Synonyms

• (way in which two things may be associated): connection, link, relationship
• (member of one's family): relative
• (act of relating a story): recounting, telling
• (mathematics: set of ordered tuples): correspondence
• See also Thesaurus:relative

Hyponyms

• (set theory): function

Derived terms

Related terms

• relate
• relational
• relative
• relator

Translations

Anagrams

• Oriental, Tirolean, oriental, taileron, tenorial

French

Etymology

From Old French relacion, from Latin rel?ti?.

Pronunciation

• IPA^(key): /??.la.sj??/

Noun

relation f (plural relations)

1. relation
2. relationship

Further reading

• “relation” in Trésor de la langue française informatisé (The Digitized Treasury of the French Language).

Anagrams

• enrôlait, oriental

Swedish

Etymology

From Latin rel?ti?.

Pronunciation

• IPA^(key): /r?la??u?n/

Noun

relation c

1. relation; how two things may be associated
2. (mathematics) relation; set of ordered tuples
3. (computing) relation; retrievable by a database

Declension

See also

• samband

Anagrams

• laotiern

English

Etymology

anti- +? symmetric.

Adjective

antisymmetric (not comparable)

1. (set theory, order theory, of a binary relation R on a set S) Having the property that, for any two distinct elements of S, at least one is not related to the other via R; equivalently, having the property that, for any x, y ? S, if both xRy and yRx then x=y.
   • 1987, David C. Buchthal, Douglas E. Cameron, Modern Abstract Algebra, Prindle, Weber & Schmidt, page 479,

     The standard example for an antisymmetric relation is the relation less than or equal to on the real number system.

   • 2006, S. C. Sharma, Metric Space, Discovery Publishing House, page 73,

     (i) The identity relation on a set A is an antisymmetric relation.

     (ii) Let R be a relation on the set N of natural numbers defined by

        x R y ${\displaystyle \Leftrightarrow }$ 'x divides y' for all x, y ? N.

     This relation is an antisymmetric relation on N.

2. (linear algebra, of certain mathematical objects) Whose sign changes on the application of a matrix transpose or some generalisation thereof:
   1. (of a matrix) Whose transpose equals its negative (i.e., M^T = ?M);
      • 1974, Robert McCredie May, Stability and Complexity in Model Ecosystems, Princeton University Press, page 193,

        The eigenvalues of an antisymmetric matrix are all purely imaginary numbers, and occur as conjugate pairs, ${\displaystyle +iw}$ and ${\displaystyle -iw}$. As a corollary it follows that an antisymmetric matrix of odd order necessarily has one eigenvalue equal to zero; antisymmetric matrices of odd order are singular.

   2. (of a tensor) That changes sign when any two indices are interchanged (e.g., T_ijk = -T_jik);
      • 1986, Millard F. Beatty Jr., Principles of Engineering Mechanics, Volume 1: Kinematics — The Geometry of Motion, Plenum Press, page 163,

        Notice that the tensors defined by:

            ${\displaystyle \textstyle T_{S}\equiv {\frac {1}{2}}(T+T^{T})}$,     ${\displaystyle \textstyle T_{A}\equiv {\frac {1}{2}}(T-T^{T}),}$     (3.47)

        are the symmetric and antisymmetric parts, respectively; they are known as the symmetric and antisymmetric parts of T.

   3. (of a bilinear form) For which B(w,v) = -B(v,w).
      • 2012, Stephanie Frank Singer, Symmetry in Mechanics: A Gentle, Modern Introduction, Springer, page 28,

        Antisymmetric bilinear forms and wedge products are defined exactly as above, only now they are functions from ${\displaystyle \mathbb {R} ^{n}\times \mathbb {R} ^{n}}$ to ${\displaystyle \mathbb {R} }$. […]

        Exercise 21 Show that every antisymmetric bilinear form on ${\displaystyle \mathbb {R} ^{3}}$ is a wedge product of two covectors.

Synonyms

• (linear algebra): skew-symmetric

Related terms

• antisymmetry
• symmetric

Translations

See also

• anticommutative
• skew-symmetric

Further reading

• Antisymmetric relation on Wikipedia.Wikipedia
• Asymmetric relation on Wikipedia.Wikipedia
• Symmetric relation on Wikipedia.Wikipedia
• Antisymmetric matrix on Wikipedia.Wikipedia
• Antisymmetric tensor on Wikipedia.Wikipedia