Definition:Total Relation
(Redirected from Definition:Dichotomous Relation)
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Definition
Let $\RR \subseteq S \times S$ be a relation on a set $S$.
Then $\RR$ is defined as total if and only if:
- $\forall a, b \in S: \tuple {a, b} \in \RR \lor \tuple {b, a} \in \RR$
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:
Some sources use the term dichotomy, but this word is also used for a concept in statistics, so will not be used in this context on $\mathsf{Pr} \infty \mathsf{fWiki}$.
Also see
- Definition:Connected Relation, a similar concept but in which it is not necessarily the case that $\forall a \in S: \tuple {a, a} \in \RR$.
- 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): Chapter $1$ Introduction: $1.7$: Terminology and Notation