nLab
symmetric relation

Definition

A (binary) relation $\sim$ on a set $A$ is symmetric if any two elements that are related in one order are also related in the other order:

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

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

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

In that case, this containment is in fact an equality.

Related concepts

• relation

  • reflexive relation

  • symmetric relation