# nLab symmetric relation

Contents

## In higher category theory

#### Graph theory

graph theory

graph

category of simple graphs

# Contents

## 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 $f \in E$, $sym(sym(f)) =_E f$

Last revised on September 22, 2022 at 18:42:01. See the history of this page for a list of all contributions to it.