# nLab antisymmetric relation

## In higher category theory

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

$\forall \left(x,y:A\right),\phantom{\rule{thickmathspace}{0ex}}x\sim y\phantom{\rule{thickmathspace}{0ex}}\wedge \phantom{\rule{thickmathspace}{0ex}}y\sim x\phantom{\rule{thickmathspace}{0ex}}⇒\phantom{\rule{thickmathspace}{0ex}}x=y$\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 $A$:

$R\cap {R}^{\mathrm{op}}\subseteq {id}_{A}$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.