Definition:Left-Total Relation
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Definition
Let $S$ and $T$ be sets.
Let $\mathcal R \subseteq S \times T$ be a relation in $S$ to $T$.
Then $\mathcal R$ is left-total iff:
- $\forall s \in S: \exists t \in T: \left({s, t}\right) \in \mathcal R$
That is, iff every element of $S$ relates to some element of $T$.
From Inverse of Right-Total is Left-Total, if $\mathcal R$ is left-total, then its inverse $\mathcal R^{-1}$ is right-total.
Alternative names
A left-total relation $\mathcal R \subseteq S \times T$ is also sometimes referred to as:
- A total relation, but this can be confused with a connected relation.
- A relation on $S$, but this can be confused with an endorelation.
Therefore the term left-total relation is usually preferred.
Multifunction
A left-total relation is also known as a multifunction, or multi-valued function.
It is usually used in this context in the field of complex analysis.
Strictly speaking, it may not actually be a function (that is, a mapping) at all, as (by implication) there may exist elements in the domain which are mapped to more than one element in the codomain.
However, if $\mathcal R$ is regarded as a function from $S$ to the power set of $T$, then left-totality of the relation is the same as totality of this lifted function. See the definition for an Induced Mapping.
