# Definition:Total Relation

## Definition

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

Then $\mathcal R$ is defined as total if and only if:

$\forall a, b \in S: \tuple {a, b} \in \mathcal R \lor \tuple {b, a} \in \mathcal R$

That is, if and only if every pair of elements is related (either or both ways round).

## Also known as

Other terms that can be found that mean the same thing as total relation are:

• dichotomy or dichotomous relation
• strictly connected relation
• complete relation.

## Also see

• Definition:Connected Relation, a similar concept but in which it is not necessarily the case that $\forall a \in S: \tuple {a, a} \in \mathcal R$.
• Results about total relations can be found here.