---
tag: 0N9SA25Z3V
kind: definition
name: Symmetric relation
msc: ["03E20"]
defines: 061J5VGSPT
uses: [08EJC6Q25H]
sources:
  - url: https://en.wikipedia.org/wiki/Symmetric_relation
    locator: lead section
---

## Definition

Let $R$ be a [[binary relation|08EJC6Q25H]] on a set $A$.
The relation $R$ is symmetric when, for all $a, b \in A$, $a \mathrel{R} b$ implies $b \mathrel{R} a$.

## Remarks

Equality is symmetric. The relation $\le$ on $\mathbb{R}$ is not symmetric, because $1 \le 2$ but not $2 \le 1$.
A relation $R$ is symmetric exactly when it is equal to its [[converse|0JCSQNNP2M]].

## Explanations

### L1 human

"Is a sibling of" is symmetric: if Ann is a sibling of Bob, then Bob is a sibling of Ann.
"Is a parent of" is not symmetric.