binary relation

English

Noun

binary relation (plural binary relations)

1. (set theory, order theory, "on" a set A) A subset of the Cartesian product A×A (the set of ordered pairs (a, b) of elements of A).
   • 1978, George Grätzer, General Lattice Theory, Academic Press, page 1:

     A partially ordered set $\langle A,\varrho \rangle$ consists of a nonvoid set $A$ and a binary relation $\varrho$ on $A$, such that $\varrho$ 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 A×B.

Usage notes

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

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 f
• Hungarian: kétváltozós reláció (hu)
• Icelandic: tvístæð vensl n pl
• Italian: relazione binaria f
• Japanese: 二項関係 (ja) (nikō-kankei)
• 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