binary relation

English

English Wikipedia has an article on:

binary relation

Wikipedia

Pronunciation

• (General American) IPA^(key): /ˈbaɪ.ner.i rɪˈleɪ.ʃən/

• Audio (General American):

  (file)

Noun

binary relation (plural binary relations)

1. (set theory, order theory, "on" a set A) A subset of the Cartesian product ${\displaystyle A\times A}$ (the set of ordered pairs (${\displaystyle a}$, ${\displaystyle b}$) of elements of ${\displaystyle A}$, alternatively written as ${\displaystyle A^{2}}$).

   Synonym: homogeneous relation

   • 1978, George Grätzer, General Lattice Theory, Academic Press, page 1:

     A partially ordered set ${\displaystyle \langle A,\varrho \rangle }$ consists of a nonvoid set ${\displaystyle A}$ and a binary relation ${\displaystyle \varrho }$ on ${\displaystyle A}$, such that ${\displaystyle \varrho }$ satisfies properties (P1)-(P3).

   • 1999, James C. Moore, Mathematical Methods for Economic Theory 1, Springer, page 24:

     1.30. Corollary. If P is a binary relation which is asymmetric and negatively transitive, then P is also transitive.
     It should be noted that a binary relation may be irreflexive and negatively transitive without being transitive; as an example, consider the standard inequality relation (≠).

   • 2005, T. S. Blyth, Lattices and Ordered Algebraic Structures, Springer, page 1:

     Definition If E is a non-empty set then by an order on E we mean a binary relation on E that is reflexive, anti-symmetric, and transitive.

2. (set theory, order theory, "on" or "between" sets A and B) A subset of the Cartesian product ${\displaystyle A\times B}$.

   Synonym: heterogeneous relation

Usage notes

• If ${\displaystyle R}$ is a relation between ${\displaystyle A}$ and ${\displaystyle B}$, then ${\displaystyle A}$ is called the domain of the relation, and ${\displaystyle B}$ is called the codomain of the relation.
• For a binary relation ${\displaystyle R}$, the notation ${\displaystyle aRb}$ signifies that ${\displaystyle (a,b)\in R}$, and one may say that ${\displaystyle a}$ is in binary relation ${\displaystyle R}$ to ${\displaystyle b}$.

Synonyms

• (order theory): correspondence, dyadic relation, 2-place relation

Hyponyms

• (order theory): dependency relation, equivalence relation

Translations

order theory

Czech: binární relace f Finnish: binäärirelaatio French: relation binaire (fr) f Hungarian: kétváltozós reláció (hu) Icelandic: tvístæð vensl n pl Italian: relazione binaria f Japanese: 二項関係 (ja) (nikō-kankei) Polish: relacja dwuargumentowa f, relacja dwuczłonowa f, relacja binarna f Romanian: relație binară (ro) f Spanish: relación binaria f

See also

• nil relation (the empty set)
• universal relation (the entire set A×A)

Further reading

• Finitary relation on Wikipedia.Wikipedia
• Order theory on Wikipedia.Wikipedia
• Binary relation on Encyclopedia of Mathematics
• Binary Relation on Wolfram MathWorld