# Asymmetric relation

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

A logically equivalent definition is 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
*a*R*b*and*b*R*a*, transitivity gives*a*R*a*, 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".

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