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.[1]
This can be written in the notation of first-order logic as
\( {\displaystyle \forall a,b\in X:aRb\rightarrow \lnot (bRa).} \)
A logically equivalent definition is \( {\displaystyle \forall a,b\in X:\lnot (aRb\wedge bRa).} \)An example of an asymmetric relation is the "less than" relation < between real numbers: if x < y, then necessarily y is not less than x. The "less than or equal" relation ≤, on the other hand, is not asymmetric, because reversing e.g. x ≤ x produces x ≤ x and both are true. Asymmetry is not the same thing as "not symmetric": the less-than-or-equal relation is an example of a relation that is neither symmetric nor asymmetric. The empty relation is the only relation that is (vacuously) both symmetric and asymmetric.
Properties
A relation is asymmetric if and only if it is both antisymmetric and irreflexive.[2]
Restrictions and converses 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.
A transitive relation is asymmetric if and only if it is irreflexive:[3] if aRb and bRa, transitivity gives aRa, contradicting irreflexivity.
As a consequence, a relation is transitive and asymmetric if and only if it is a strict partial order.
Not all asymmetric relations are strict partial orders. An example of an asymmetric non-transitive, even antitransitive relation is the rock paper scissors relation: if X beats Y, then Y does not beat X; and if X beats Y and Y beats Z, then X does not beat Z.
An asymmetric relation need not have the connex property. For example, the strict subset relation ⊊ is asymmetric, and neither of the sets {1,2} and {3,4} is a strict subset of the other.
See also
Tarski's axiomatization of the reals – part of this is the requirement that < over the real numbers be asymmetric.
References
Gries, David; Schneider, Fred B. (1993), A Logical Approach to Discrete Math, Springer-Verlag, p. 273.
Nievergelt, Yves (2002), Foundations of Logic and Mathematics: Applications to Computer Science and Cryptography, Springer-Verlag, p. 158.
Flaška, V.; Ježek, J.; Kepka, T.; Kortelainen, J. (2007). Transitive Closures of Binary Relations I (PDF). Prague: School of Mathematics - Physics Charles University. p. 1. Archived from the original (PDF) on 2013-11-02. Retrieved 2013-08-20. Lemma 1.1 (iv). Note that this source refers to asymmetric relations as "strictly antisymmetric".
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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