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 $a = b$

• Relation is Connected and Reflexive iff Total

• Definition:Trichotomy

• 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): connected (of a relation)