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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   consists of a nonvoid set   and a binary relation   on  , such that   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 notesEdit

For a binary relation  , the notation   signifies that  , and one may say that   is in binary relation   to  .

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