# Definition:Left-Total Relation

Jump to navigation
Jump to search

## Definition

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

Then $\RR$ is **left-total** if and only if:

- $\forall s \in S: \exists t \in T: \tuple {s, t} \in \RR$

That is, if and only if 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 $\mathsf{Pr} \infty \mathsf{fWiki}$ already has a definition for such a concept - 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.

## Multifunction

A **multifunction** is a **left-total relation** $\RR$ which is specifically not many-to-one or one-to-one.

That is, for each element $s$ of the domain of $\RR$, there exists more than one $t$ in the codomain of $\RR$ such that $\tuple {s, t} \in \RR$.

## Also see

- Definition:Serial Relation: an endorelation $\RR: S \to S$ which is
**left-total**

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

## Sources

- 1968: A.N. Kolmogorov and S.V. Fomin:
*Introductory Real Analysis*... (previous) ... (next): $\S 1.4$: Decomposition of a set into classes. Equivalence relations - 1972: Murray R. Spiegel and R.W. Boxer:
*Theory and Problems of Statistics*(SI ed.) ... (previous) ... (next): Chapter $1$: Functions