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.

Relation to graphs

A set with a symmetric relation is the same as a loop digraph $(V, E, s:E \to V, t:E \to V)$ with a function $sym:E \to E$ such that

• for every $f \in E$, $s(f) =_V t(sym(f))$
• for every $f \in E$, $t(f) =_V s(sym(f))$
• for every $f \in E$, $sym(sym(f)) =_E f$

Related concepts

• relation

  • reflexive relation

  • symmetric relation

• loop graph object

• equivalence relation