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 iff:
 * $\forall a, b \in S: \left({a, b}\right) \in \mathcal R \lor \left({b, a}\right) \in \mathcal R$

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

From Relation Connected and Reflexive iff Total‎ it is seen that a total relation is a connected relation which is also reflexive.

Also known as
Some sources call this a dichotomy or dichotomous relation.

Also see

 * Connected Relation, a similar concept but in which it is not necessarily the case that $\forall a \in S: \left({a, a}\right) \in \mathcal R$.


 * Trichotomy


 * Left-Total Relation and Right-Total Relation, which are in fact different concepts.