Asymmetric relation

Examples

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 ${\displaystyle ⊊}$ 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 ${\displaystyle \le }$. This is not asymmetric, because reversing for example, ${\displaystyle x\le x}$ produces ${\displaystyle x\le x}$ 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".

The empty relation is the only relation that is (vacuously) both symmetric and asymmetric.

Properties

The following conditions are sufficient for a relation ${\displaystyle R}$ to be asymmetric:

• ${\displaystyle R}$ is irreflexive and anti-symmetric (this is also necessary)
• ${\displaystyle R}$ is irreflexive and transitive. A transitive relation is asymmetric if and only if it is irreflexive: if ${\displaystyle aRb}$ and ${\displaystyle bRa,}$ transitivity gives ${\displaystyle aRa,}$ contradicting irreflexivity. Such a relation is a strict partial order.
• ${\displaystyle R}$ is irreflexive and satisfies semiorder property 1 (there do not exist two mutually incomparable two-point linear orders)
• ${\displaystyle R}$ is antitransitive and anti-symmetric
• ${\displaystyle R}$ is antitransitive and transitive
• ${\displaystyle R}$ is antitransitive and satisfies semi-order property 1