Definition:Total Relation

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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.


Sources