Asymmetric relation

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Asymmetric often means, simply: not symmetric. In this sense an asymmetric relation is a relation, in particular 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 this sense if, 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).

Being asymmetric in this sense is the same as being both antisymmetric and irreflexive.

Asymmetry in the second sense implies asymmetry in the first sense, but the reverse implication does not hold. Empty relations are, vacuously, both symmetric and asymmetric (in the second sense only).

See also Symmetry in mathematics.

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