# Definition:Connected Relation

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*This page is about Connected Relation in the context of Relation Theory. For other uses, see 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:

- $\forall a, b \in S: a \ne b \implies \tuple {a, b} \in \mathcal R \lor \tuple {b, a} \in \mathcal R$

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 $\tuple {a, b} \in \mathcal R \lor \tuple {b, a} \in \mathcal R$ 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): $\S 1$: Naive Set Theory: $\S 1.5$: Relations - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**connected**(of a relation)