# Definition:Connected Relation

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

## Definition

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

Then $\RR$ is connected if and only if:

$\forall a, b \in S: a \ne b \implies \tuple {a, b} \in \RR \lor \tuple {b, a} \in \RR$

That is, if and only if every pair of distinct elements is comparable.

## 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.

A set on which $\RR$ is connected can be referred to as an $\RR$-connected set.

## Also see

• Definition:Total Relation: a connected relation which also insists that $\tuple {a, b} \in \RR \lor \tuple {b, a} \in \RR$ even for $a = b$
• Results about connected relations can be found here.