|
In mathematics, a binary relation ''R'' over a set ''X'' is transitive if whenever an element ''a'' is related to an element ''b'', and ''b'' is in turn related to an element ''c'', then ''a'' is also related to ''c''. Transitivity is a key property of both partial order relations and equivalence relations. == Formal definition == In terms of set theory, the transitive relation can be defined as: : ==Examples== For example, "is greater than", "is at least as great as," and "is equal to" (equality) are transitive relations: : whenever A > B and B > C, then also A > C : whenever A ≥ B and B ≥ C, then also A ≥ C : whenever A = B and B = C, then also A = C. On the other hand, "is the mother of" is not a transitive relation, because if Alice is the mother of Brenda, and Brenda is the mother of Claire, then Alice is not the mother of Claire. What is more, it is antitransitive: Alice can ''never'' be the mother of Claire. Then again, in biology we often need to consider motherhood over an arbitrary number of generations: the relation "is a matrilinear ancestor of". This ''is'' a transitive relation. More precisely, it is the transitive closure of the relation "is the mother of". More examples of transitive relations: * "is a subset of" (set inclusion) * "divides" (divisibility) * "implies" (implication) 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「Transitive relation」の詳細全文を読む スポンサード リンク
|