Definition:Left-Total Relation

Definition
Let $S$ and $T$ be sets.

Let $\RR \subseteq S \times T$ be a relation in $S$ to $T$.

Then $\RR$ is left-total :
 * $\forall s \in S: \exists t \in T: \tuple {s, t} \in \RR$

That is, every element of $S$ relates to some element of $T$.

Also known as
A left-total relation $\RR \subseteq S \times T$ 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
 * A multiple-valued function or multifunction, but the latter is usually reserved for complex functions

The term left-total relation is usually preferred.

Also see

 * Definition:Right-Total Relation
 * Inverse of Left-Total Relation is Right-Total


 * Definition:Serial Relation: a relation $\RR: S \to S$ which is '''left-total