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 $a R b$ if and only if $(a, b) \in R,$ which means that $a R b$ is shorthand for $(a, b) \in R.$ The expression $a R b$ is read as "$a$ is related to $b$ by $R.$" The binary relation $R$ is called if for all $a, b \in X,$ if $a R b$ is true then $b R a$ 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
$\forall a, b \in X: a R b \implies \lnot(b R a).$

A logically equivalent definition is:

:for all $a, b \in X,$ at least one of $a R b$ and $b R a$ is ,
which in first-order logic can be written as:
$\forall a, b \in X: \lnot(a R b \wedge b R a).$

An example of an asymmetric relation is the "less than" relation $\,<\,$ between real numbers: if $x < y$ 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: Lemma 1.1 (iv). Note that this source refers to asymmetric relations as "strictly antisymmetric". 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 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 $\$ and $\$ is a strict subset of the other. A relation is connex if and only if its complement is asymmetric.

See also

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

References

{{DEFAULTSORT:Asymmetric Relation
Binary relations
Asymmetry