Definition:Connected Relation
From ProofWiki
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).
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.
