|
In mathematics an asymmetric relation is a binary relation on a set ''X'' where: *For all ''a'' and ''b'' in ''X'', if ''a'' is related to ''b'', then ''b'' is not related to ''a''.〔.〕 In mathematical notation, this is: :. __NOTOC__ == Examples == An example is < (less-than): if x < y, then necessarily y is not less than x. In fact, one of Tarski's axioms characterizing the real numbers R is that < over R is asymmetric. An asymmetric relation need not be total. For example, strict subset or ⊊ is asymmetric, and neither of the sets and is a strict subset of the other. In general, every strict partial order is asymmetric, and conversely, every transitive asymmetric relation is a strict partial order. Not all asymmetric relations are strict partial orders, however. An example of an asymmetric intransitive relation is the rock-paper-scissors relation: if X beats Y, then Y does not beat X, but no one choice wins all the time. The ≤ (less than or equal) operator, on the other hand, is not asymmetric, because reversing x ≤ x produces x ≤ x and both are true. In general, any relation in which ''x'' R ''x'' holds for some ''x'' (that is, which is not irreflexive) is also not asymmetric. Asymmetric is not the same thing as "not symmetric": a relation can be neither symmetric nor asymmetric, such as ≤, or can be both, only in the case of the empty relation (vacuously). 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「asymmetric relation」の詳細全文を読む スポンサード リンク
|