Definition:Connected Relation

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

Then $\mathcal R$ is connected :
 * $\forall a, b \in S: a \ne b \implies \tuple {a, b} \in \mathcal R \lor \tuple {b, a} \in \mathcal R$

That is, 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 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 $a = b$


 * Relation is Connected and Reflexive iff Total


 * Definition:Trichotomy