See binary relation in All languages combined, or Wiktionary
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Moore, Mathematical Methods for Economic Theory 1, Springer, page 24:", "text": "1.30. Corollary. If P is a binary relation which is asymmetric and negatively transitive, then P is also transitive.\nIt should be noted that a binary relation may be irreflexive and negatively transitive without being transitive; as an example, consider the standard inequality relation (≠).", "type": "quote" }, { "ref": "2005, T. S. Blyth, Lattices and Ordered Algebraic Structures, Springer, page 1:", "text": "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.", "type": "quote" } ], "glosses": [ "A subset of the Cartesian product A×A (the set of ordered pairs (a, b) of elements of A)." ], "id": "en-binary_relation-en-noun-QVsM9Asj", "links": [ [ "set theory", "set theory" ], [ "subset", "subset" ], [ "Cartesian product", "Cartesian product" ], [ "ordered pair", "ordered pair" ], [ "element", "element" ] ], "qualifier": "\"on\" a set A", "raw_glosses": [ "(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)." ], "topics": [ "mathematics", "order-theory", "sciences", "set-theory" ] }, { "categories": [ { "kind": "topical", "langcode": "en", "name": "Set theory", "orig": "en:Set theory", "parents": [ "Mathematics", "Formal sciences", "Sciences", "All topics", "Fundamental" ], "source": "w" } ], "glosses": [ "A subset of the Cartesian product A×B." ], "id": "en-binary_relation-en-noun-L9v2tqxj", "links": [ [ "set theory", "set theory" ], [ "subset", "subset" ], [ "Cartesian product", "Cartesian product" ] ], "qualifier": "\"on\" or \"between\" sets A and B", "raw_glosses": [ "(set theory, order theory, \"on\" or \"between\" sets A and B) A subset of the Cartesian product A×B." ], "topics": [ "mathematics", "order-theory", "sciences", "set-theory" ] } ], "synonyms": [ { "_dis1": "0 0", "topics": [ "order-theory", "mathematics", "sciences" ], "word": "correspondence" }, { "_dis1": "0 0", "topics": [ "order-theory", "mathematics", "sciences" ], "word": "dyadic relation" }, { "_dis1": "0 0", "topics": [ "order-theory", "mathematics", "sciences" ], "word": "2-place relation" } ], "translations": [ { "_dis1": "50 50", "code": "cs", "lang": "Czech", "sense": "order theory", "tags": [ "feminine" ], "word": "binární relace" }, { "_dis1": "50 50", "code": "fi", "lang": "Finnish", "sense": "order theory", "word": "binäärirelaatio" }, { "_dis1": "50 50", "code": "fr", "lang": "French", "sense": "order theory", "tags": [ "feminine" ], "word": "relation binaire" }, { "_dis1": "50 50", "code": "hu", "lang": "Hungarian", "sense": "order theory", "word": "kétváltozós reláció" }, { "_dis1": "50 50", "code": "is", "lang": "Icelandic", "sense": "order theory", "tags": [ "neuter", "plural" ], "word": "tvístæð vensl" }, { "_dis1": "50 50", "code": "it", "lang": "Italian", "sense": "order theory", "tags": [ "feminine" ], "word": "relazione binaria" }, { "_dis1": "50 50", "code": "ja", "lang": "Japanese", "roman": "nikō-kankei", "sense": "order theory", "word": "二項関係" }, { "_dis1": "50 50", "code": "ro", "lang": "Romanian", "sense": "order theory", "tags": [ "feminine" ], "word": "relație binară" }, { "_dis1": "50 50", "code": "es", "lang": "Spanish", "sense": "order theory", "tags": [ "feminine" ], "word": "relación binaria" } ], "wikipedia": [ "binary relation" ], "word": "binary relation" }
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