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In mathematics
Asymmetric Relation

An asymmetric relation is a binary relation $$R$$ defined on a set of elements such that if $$a R b$$ holds for elements $$a$$ and $$b$$, then $$b R a$$ must be false. Stated differently, an asymmetric relation is characterized by a necessary absence of symmetry of the relation in the opposite direction.

Inequalities exemplify an asymmetric relation. Consider elements $$a$$ and $$b$$. If $$a$$ is less than $$b$$ ($$a < b$$), then $$a$$ cannot be greater than $$b$$ ($$a \ngtr b$$). This highlights how the relations "less than", and similarly "greater than", are not symmetric.

In contrast, if $$a$$ is equal to $$b$$ ($$a = b$$), then $$b$$ is also equal to $$a$$ ($$b = a$$). Thus the binary relation "equal to" is a symmetric one. (See also: Equivalence relation)

In graph theory, relations can be represented by graphs, wherein directed edges represent asymmetric relations.

Asymmetric Graphs