# Definition:Connected Relation

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

## 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.