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| In [[mathematics]] an '''asymmetric relation''' is a [[binary relation]] on a set ''X'' where:
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| *For all ''a'' and ''b'' in ''X'', if ''a'' is related to ''b'', then ''b'' is not related to ''a''.<ref>{{citation|first1=David|last1=Gries|author1-link=David Gries|first2=Fred B.|last2=Schneider|author2-link=Fred B. Schneider|title=A Logical Approach to Discrete Math|publisher=Springer-Verlag|year=1993|page=[http://books.google.com/books?id=ZWTDQ6H6gsUC&pg=PA273 273]}}.</ref>
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| In mathematical notation, this is:
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| :<math>\forall a, b \in X,\ a R b \; \Rightarrow \lnot(b R a)</math>.
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| __NOTOC__
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| == Examples ==
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| An example is < ([[Inequality (mathematics)|less-than]]): if x < y, then necessarily y is not less than x. In fact, one of [[Tarski's axiomatization of the reals|Tarski's axioms characterizing the real numbers '''R''']] is that < over '''R''' is asymmetric.
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| An asymmetric relation need not be [[total relation|total]]. For example, [[strict subset]] or ⊊ is asymmetric, and neither of the sets {1,2} and {3,4} is a strict subset of the other. In general, every [[strict partial order]] is asymmetric, and conversely, every [[transitive relation|transitive]] asymmetric relation is a strict partial order.
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| Not all asymmetric relations are strict partial orders, however. An example of an asymmetric [[Intransitivity|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.
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| 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 [[reflexive relation|irreflexive]]) is also not asymmetric.
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| Asymmetric is not the same thing as "not [[symmetric relation|symmetric]]": a relation can be neither symmetric nor asymmetric, such as ≤, or can be both, only in the case of the empty relation ([[vacuous truth|vacuously]]).
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| == Properties==
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| * A relation is asymmetric if and only if it is both [[antisymmetric relation|antisymmetric]] and [[reflexive relation|irreflexive]].<ref>{{citation|first1=Yves|last1=Nievergelt|title=Foundations of Logic and Mathematics: Applications to Computer Science and Cryptography|publisher=Springer-Verlag|year=2002|page=[http://books.google.com/books?id=_H_nJdagqL8C&pg=PA158 158]}}.</ref>
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| * [[Binary_relation#Restriction|Restrictions]] and [[inverse relation|inverses]] of asymmetric relations are also asymmetric. For example, the restriction of < from the reals to the integers is still asymmetric, and the inverse > of < is also asymmetric.
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| * A [[transitive relation]] is asymmetric if and only if it is irreflexive:<ref>{{cite book|last1=Flaška|first1=V.|last2=Ježek|first2=J.|last3=Kepka|first3=T.|last4=Kortelainen|first4=J.|title=Transitive Closures of Binary Relations I|year=2007|publisher=School of Mathematics - Physics Charles University|location=Prague|page=1|url=http://www.karlin.mff.cuni.cz/~jezek/120/transitive1.pdf}} Lemma 1.1 (iv). Note that this source refers to asymmetric relations as "strictly antisymmetric".</ref> if ''a'' R ''b'' and ''b'' R ''a'', transitivity gives ''a'' R ''a'', contradicting irreflexivity.
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| ==See also==
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| *[[Symmetry in mathematics]]
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| *[[Symmetry]]
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| *[[Antisymmetric relation]]
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| ==References==
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| {{Reflist}}
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| {{DEFAULTSORT:Asymmetric Relation}}
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| [[Category:Mathematical relations]]
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