binary relation

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binary relation

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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.

• 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 ${\displaystyle b}$.

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

• (order theory): dependency relation, equivalence relation

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

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