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 ${\displaystyle a.}$

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:

which in first-order logic can be written as: $${\displaystyle \forall a,b\in X:\lnot (aRb\wedge bRa).}$$ A relation is asymmetric if and only if it is both antisymmetric and irreflexive, so this may also be taken as a definition.

An example of an asymmetric relation is the "less than" relation ${\displaystyle \,<\,}$ between real numbers: if ${\displaystyle x<y}$ then necessarily ${\displaystyle y}$ is not less than ${\displaystyle x.}$ More generally, any strict partial order is an asymmetric relation. 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 ${\displaystyle Z.}$

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 converse or dual ${\displaystyle \,>\,}$ of ${\displaystyle \,<\,}$ is also asymmetric.

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.

A non-example is the "less than or equal" relation ${\displaystyle \leq }$. This is not asymmetric, because reversing for example, ${\displaystyle x\leq x}$ produces ${\displaystyle x\leq x}$ and both are true. The less-than-or-equal relation is an example of a relation that is neither symmetric nor asymmetric, showing that asymmetry is not the same thing as "not symmetric".