# Asymmetric relation

In mathematics, an asymmetric relation is a binary relation $R$ on a set $X$ where for all $a,b\in X,$ if $a$ is related to $b$ then $b$ is not related to $a.$ ## Formal definition

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

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

A logically equivalent definition is:

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

which in first-order logic can be written as:

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

An example of an asymmetric relation is the "less than" relation $\,<\,$  between real numbers: if $x  then necessarily $y$  is not less than $x.$  The "less than or equal" relation $\,\leq ,$  on the other hand, is not asymmetric, because reversing for example, $x\leq x$  produces $x\leq x$  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.
• 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: 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 $\,\subsetneq \,$  is asymmetric, and neither of the sets $\{1,2\}$  and $\{3,4\}$ is a strict subset of the other. A relation is connex if and only if its complement is asymmetric.

• Tarski's axiomatization of the reals – part of this is the requirement that $\,<\,$  over the real numbers be asymmetric.