Definition:Connected Relation

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

Then $\mathcal R$ is defined as connected iff:
 * $\forall \left({a, b}\right) \in \mathcal R: a \ne b \implies \left({a, b}\right) \in \mathcal R \lor \left({b, a}\right) \in \mathcal R$

That is, iff every pair of distinct elements is related (either or both ways round).

Also known as
This can also be called a total relation but beware of confusing this with left-total and right-total relations, which mean something else altogether.

Some sources call this a dichotomy or dichotomous relation.

Also see

 * Trichotomy