# Definition:Connected Relation

*This page is about connected relations. For other uses, see Definition:Connected.*

## Contents

## Definition

Let $\mathcal R \subseteq S \times S$ be a relation on a set $S$.

Then $\mathcal R$ is **connected** if and only if:

- $\forall a, b \in S: a \ne b \implies \left({a, b}\right) \in \mathcal R \lor \left({b, a}\right) \in \mathcal R$

That is, if and only if every pair of distinct elements is related (either or both ways round).

## Also known as

Some sources use the term **weakly connected**, using the term **strictly connected relation** for what is defined on $\mathsf{Pr} \infty \mathsf{fWiki}$ as total relation.

## Also see

- Definition:Total Relation: a
**connected relation**which also insists that $\left({a, b}\right) \in \mathcal R \lor \left({b, a}\right) \in \mathcal R$ even for $a = b$

- Results about
**connected relations**can be found here.

## Sources

- 1993: Keith Devlin:
*The Joy of Sets: Fundamentals of Contemporary Set Theory*(2nd ed.) ... (previous) ... (next): $\S 1.5$: Relations