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

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

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

- Results about
**total relations**can be found here.

## Sources

- 1960: Paul R. Halmos:
*Naive Set Theory*... (previous) ... (next): $\S 14$: Order - 1994: Martin J. Osborne and Ariel Rubinstein:
*A Course in Game Theory*... (previous) ... (next): $1.7$: Terminology and Notation