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In mathematics, a binary relation on a set ''A'' is a collection of ordered pairs of elements of ''A''. In other words, it is a subset of the Cartesian product ''A''2 = . More generally, a binary relation between two sets ''A'' and ''B'' is a subset of . The terms correspondence, dyadic relation and 2-place relation are synonyms for binary relation. An example is the "divides" relation between the set of prime numbers P and the set of integers Z, in which every prime ''p'' is associated with every integer ''z'' that is a multiple of ''p'' (but with no integer that is not a multiple of ''p''). In this relation, for instance, the prime 2 is associated with numbers that include −4, 0, 6, 10, but not 1 or 9; and the prime 3 is associated with numbers that include 0, 6, and 9, but not 4 or 13. Binary relations are used in many branches of mathematics to model concepts like "is greater than", "is equal to", and "divides" in arithmetic, "is congruent to" in geometry, "is adjacent to" in graph theory, "is orthogonal to" in linear algebra and many more. The concept of function is defined as a special kind of binary relation. Binary relations are also heavily used in computer science. A binary relation is the special case of an ''n''-ary relation ''R'' ⊆ ''A''1 × … × ''A''''n'', that is, a set of ''n''-tuples where the ''j''th component of each ''n''-tuple is taken from the ''j''th domain ''A''''j'' of the relation. An example for a ternary relation on Z×Z×Z is "lies between ... and ...", containing e.g. the triples , , and . In some systems of axiomatic set theory, relations are extended to classes, which are generalizations of sets. This extension is needed for, among other things, modeling the concepts of "is an element of" or "is a subset of" in set theory, without running into logical inconsistencies such as Russell's paradox. ==Formal definition== A binary relation ''R'' is usually defined as an ordered triple (''X'', ''Y'', ''G'') where ''X'' and ''Y'' are arbitrary sets (or classes), and ''G'' is a subset of the Cartesian product ''X'' × ''Y''. The sets ''X'' and ''Y'' are called the domain (or the set of departure) and codomain (or the set of destination), respectively, of the relation, and ''G'' is called its graph. The statement (''x'',''y'') ∈ ''G'' is read "''x'' is ''R''-related to ''y''", and is denoted by ''xRy'' or ''R''(''x'',''y''). The latter notation corresponds to viewing ''R'' as the characteristic function on ''X'' × ''Y'' for the set of pairs of ''G''. The order of the elements in each pair of ''G'' is important: if ''a'' ≠ ''b'', then ''aRb'' and ''bRa'' can be true or false, independently of each other. Resuming the above example, the prime 3 divides the integer 9, but 9 doesn't divide 3. A relation as defined by the triple (''X'', ''Y'', ''G'') is sometimes referred to as a correspondence instead. In this case the relation from ''X'' to ''Y'' is the subset ''G'' of ''X'' × ''Y'', and "from ''X'' to ''Y''" must always be either specified or implied by the context when referring to the relation. In practice correspondence and relation tend to be used interchangeably. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Binary relation」の詳細全文を読む スポンサード リンク
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