Asymmetric often means, simply: not symmetric. In this sense an asymmetric relation is a binary relation which is not a symmetric relation.
In some texts the word is given the following stronger definition. A relation R on X is asymmetric in the following sense.
- 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:
.
In this sense, a relation is asymmetric if and only if it is both antisymmetric and irreflexive.
For a transitive relation, asymmetry is equivalent to irreflexivity.
For nonempty relations, asymmetry in the second sense implies asymmetry in the first sense, but the reverse implication does not hold. Empty relations are, vacuously, both asymmetric (in the second sense only) and symmetric.
See also
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