nLab antisymmetric relation

A (binary) relation $\sim$ on a set $A$ is antisymmetric if any two elements that are related in both orders are equal:

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

In the language of the $2$-poset-with-duals Rel of sets and relations, a relation $R: A \to A$ is antisymmetric if its intersection with its reverse is contained in the identity relation on $A$:

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

If an antisymmetric relation is also reflexive (as most are in practice), then this containment becomes an equality.

See also

• internal antisymmetric relation