# nLab asymmetric relation

## In higher category theory

A (binary) relation $\sim$ on a set $A$ is asymmetric if no two elements are related in both orders:

$\forall (x, y: A),\; x \sim y \;\Rightarrow\; y \nsim x$

In the language of the $2$-poset-with-duals Rel of sets and relations, a relation $R: A \to A$ is asymmetric if it is disjoint from its dual:

$R \cap R^{op} \subseteq \empty$

Of course, this containment is in fact an equality.

An asymmetric relation is necessarily irreflexive.

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