Definition:Asymmetric Relation
Definition
Let $\RR \subseteq S \times S$ be a relation in $S$.
Definition 1
$\RR$ is asymmetric if and only if:
- $\tuple {x, y} \in \RR \implies \tuple {y, x} \notin \RR$
That is:
- $x \mathop \RR y \implies \neg \paren {y \mathop \RR x}$
Definition 2
$\RR$ is asymmetric if and only if it and its inverse are disjoint:
- $\RR \cap \RR^{-1} = \O$
Antisymmetric and Asymmetric Relations
Note the difference between:
- An asymmetric relation, in which the fact that $\tuple {x, y} \in \RR$ means that $\tuple {y, x}$ is definitely not in $\RR$
and:
- An antisymmetric relation, in which there may be instances of both $\tuple {x, y} \in \RR$ and $\tuple {y, x} \in \RR$ but if there are, then it means that $x$ and $y$ have to be the same object.
Also defined as
Some sources (possibly erroneously or carelessly) gloss over the differences between this and the definition for an antisymmetric relation, and end up using a definition for antisymmetric which comes too close to one for asymmetric.
An example is 1964: Steven A. Gaal: Point Set Topology:
- [After having discussed antireflexivity] ... antisymmetry expresses the additional fact that at most one of the possibilities $a \mathrel \RR b$ or $b \mathrel \RR a$ can take place.
Some sources specifically define a relation as anti-symmetric what has been defined on $\mathsf{Pr} \infty \mathsf{fWiki}$ as asymmetric.
From 1955: John L. Kelley: General Topology: Chapter $0$: Relations:
- ... the relation $R$ is anti-symmetric iff it is never the case that both $x R y$ and $y R x$.
Examples
Parent
Asymmetric Relation/Examples/Parent
Greater Than
The relation $\RR$ defined on the set of real numbers $\R$:
- $\forall x, y \in \R: x \mathop \RR y \iff x > y$
where:
- $>$ denotes the conventional greater than relation
is an asymmetric relation.
Less Than
The relation $\RR$ defined on the set of real numbers $\R$:
- $\forall x, y \in \R: x \mathop \RR y \iff x < y$
where:
- $<$ denotes the conventional less than relation
is an asymmetric relation.
Also see
- Results about asymmetric relations can be found here.
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): asymmetric relation
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): asymmetric relation