# 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.