Definition:Total Relation
(Redirected from Definition:Strictly Connected 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 statistcis, 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 $\forall a \in S: \tuple {a, a} \in \RR$.
- 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