# 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]}

In mathematical notation, this is:

An example 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 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.

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 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.
- 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. - An asymmetric relation need not be 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 asymmetric relation is a strict partial order. Not all asymmetric relations are strict partial orders. An example of an asymmetric intransitive relation is the rock-paper-scissors relation: if X beats Y, then Y does not beat X; but if X beats Y and Y beats Z, then X does not beat Z.

## 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. Lemma 1.1 (iv). Note that this source refers to asymmetric relations as "strictly antisymmetric".

This article is issued from Wikipedia - version of the 10/27/2016. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.