# binary relation

## English

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### 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 ${\displaystyle \langle A,\varrho \rangle }$  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 A×B.

#### Usage notes

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 ${\displaystyle b}$ .