Antisymmetric relation

All definitions tacitly require the homogeneous relation

{\displaystyle aRc.}

A term's definition may require additional properties that are not listed in this table.

In mathematics, a binary relation

is antisymmetric if there is no pair of distinct elements of

is antisymmetric precisely if for all

must not hold

{\displaystyle {\text{if }}\,aRb\,{\text{ with }}\,a\neq b\,{\text{ then }}\,bRa\,{\text{ must not hold}},}

or equivalently,

The definition of antisymmetry says nothing about whether

{\displaystyle aRa}

actually holds or not for any

An antisymmetric relation

), irreflexive (that is,

), or neither reflexive nor irreflexive.

A relation is asymmetric if and only if it is both antisymmetric and irreflexive.

The divisibility relation on the natural numbers is an important example of an antisymmetric relation.

In this context, antisymmetry means that the only way each of two numbers can be divisible by the other is if the two are, in fact, the same number; equivalently, if

are distinct and

The usual order relation

on the real numbers is antisymmetric: if for two real numbers

Similarly, the subset order

on the subsets of any given set is antisymmetric: given two sets

must contain all the same elements and therefore be equal:

A real-life example of a relation that is typically antisymmetric is "paid the restaurant bill of" (understood as restricted to a given occasion).

Typically, some people pay their own bills, while others pay for their spouses or friends.

As long as no two people pay each other's bills, the relation is antisymmetric.

Partial and total orders are antisymmetric by definition.

A relation can be both symmetric and antisymmetric (in this case, it must be coreflexive), and there are relations which are neither symmetric nor antisymmetric (for example, the "preys on" relation on biological species).

Antisymmetry is different from asymmetry: a relation is asymmetric if and only if it is antisymmetric and irreflexive.