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

That is, 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 $\forall a \in S: \tuple {a, a} \in \mathcal R$.


 * Relation is Connected and Reflexive iff Total: a total relation is a connected relation which is also reflexive.


 * Definition:Trichotomy


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