Asymmetric relation

Asymmetric often means, simply: not symmetric. In this sense an asymmetric relation is a binary relation which is not a symmetric relation.

That is,

$\lnot(\forall a, b \in X,\ a R b \; \Rightarrow b R a)$.

or equivalently,

$\exists a, b \in X,\ a R b \; \land \; \lnot(b R a)$.

In some texts the word is given the following stronger definition:

• For all a and b in X, if a is related to b, then b is not related to a.

In mathematical notation, this is:

$\forall a, b \in X,\ a R b \; \Rightarrow \lnot(b R a)$.

In this sense, a relation is asymmetric if and only if it is both antisymmetric and irreflexive.

For nonempty relations, asymmetry in the second definition given here implies asymmetry in the first sense, but the reverse does not hold. Empty relations are, vacuously, both asymmetric (in the second sense only) and symmetric.

See also

• Symmetry in mathematics
• Symmetry
• Antisymmetric relation