Definition:Asymmetric Relation

Definition
Let $\mathcal R \subseteq S \times S$ be a relation in $S$.

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 :


 * [After having discussed antireflexivity] ... antisymmetry expresses the additional fact that at most one of the possibilities $a \mathop {\mathcal R} b$ or $b \mathop {\mathcal R} a$ can take place.

Some sources specifically define a relation as anti-symmetric what has been defined on as asymmetric

From : Chapter $0$: Relations:
 * ... the relation $R$ is anti-symmetric iff it is never the case that both $x R y$ and $y R x$.

Also see

 * Equivalence of Definitions of Asymmetric Relation


 * Definition:Symmetry (Relation)


 * Definition:Symmetric Relation
 * Definition:Antisymmetric Relation
 * Definition:Non-symmetric Relation