---
tag: 0XEGNW851S
kind: definition
name: Reflexive relation
msc: ["03E20"]
defines: 0X8NMST0J2
uses: [08EJC6Q25H]
sources:
  - url: https://en.wikipedia.org/wiki/Reflexive_relation
    locator: lead section
---

## Definition

Let $R$ be a [[binary relation|08EJC6Q25H]] on a set $A$.
The relation $R$ is reflexive when $a \mathrel{R} a$ for each $a \in A$.

## Remarks

On $\mathbb{R}$, the relation $\le$ is reflexive and the relation $<$ is not.
A relation is irreflexive when no element is related to itself.
Reflexivity is one of the conditions for an [[equivalence relation|0SGTMBEWRF]] and for a [[partial order|011C29Q1S1]].

## Explanations

### L1 human

In the graph of a reflexive relation, each point has a loop to itself.
"Has the same birthday as" is reflexive, because each person has the same birthday as themselves.