# Asymmetric relation

In mathematics, an asymmetric relation is a binary relation ${\displaystyle R}$ on a set ${\displaystyle X}$ where for all ${\displaystyle a,b\in X,}$ if ${\displaystyle a}$ is related to ${\displaystyle b}$ then ${\displaystyle b}$ is not related to ${\displaystyle a.}$[1]

## Formal definition

A binary relation on ${\displaystyle X}$  is any subset ${\displaystyle R}$  of ${\displaystyle X\times X.}$  Given ${\displaystyle a,b\in X,}$  write ${\displaystyle aRb}$  if and only if ${\displaystyle (a,b)\in R,}$  which means that ${\displaystyle aRb}$  is shorthand for ${\displaystyle (a,b)\in R.}$  The expression ${\displaystyle aRb}$  is read as "${\displaystyle a}$  is related to ${\displaystyle b}$  by ${\displaystyle R.}$ " The binary relation ${\displaystyle R}$  is called asymmetric if for all ${\displaystyle a,b\in X,}$  if ${\displaystyle aRb}$  is true then ${\displaystyle bRa}$  is false; that is, if ${\displaystyle (a,b)\in R}$  then ${\displaystyle (b,a)\not \in R.}$  This can be written in the notation of first-order logic as

${\displaystyle \forall a,b\in X:aRb\implies \lnot (bRa).}$

A logically equivalent definition is:

for all ${\displaystyle a,b\in X,}$  at least one of ${\displaystyle aRb}$  and ${\displaystyle bRa}$  is false,

which in first-order logic can be written as:

${\displaystyle \forall a,b\in X:\lnot (aRb\wedge bRa).}$

An example of an asymmetric relation is the "less than" relation ${\displaystyle \,<\,}$  between real numbers: if ${\displaystyle x  then necessarily ${\displaystyle y}$  is not less than ${\displaystyle x.}$  The "less than or equal" relation ${\displaystyle \,\leq ,}$  on the other hand, is not asymmetric, because reversing for example, ${\displaystyle x\leq x}$  produces ${\displaystyle x\leq x}$  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 ${\displaystyle \,<\,}$  from the reals to the integers is still asymmetric, and the inverse ${\displaystyle \,>\,}$  of ${\displaystyle \,<\,}$  is also asymmetric.
• A transitive relation is asymmetric if and only if it is irreflexive:[3] if ${\displaystyle aRb}$  and ${\displaystyle bRa,}$  transitivity gives ${\displaystyle aRa,}$  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 ${\displaystyle X}$  beats ${\displaystyle Y,}$  then ${\displaystyle Y}$  does not beat ${\displaystyle X;}$  and if ${\displaystyle X}$  beats ${\displaystyle Y}$  and ${\displaystyle Y}$  beats ${\displaystyle Z,}$  then ${\displaystyle X}$  does not beat ${\displaystyle Z.}$
• An asymmetric relation need not have the connex property. For example, the strict subset relation ${\displaystyle \,\subsetneq \,}$  is asymmetric, and neither of the sets ${\displaystyle \{1,2\}}$  and ${\displaystyle \{3,4\}}$ is a strict subset of the other. A relation is connex if and only if its complement is asymmetric.