# Definition:Left-Total Relation/Multifunction

## Definition

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 $\tuple {s, t} \in \RR$.

Hence a **multifunction** is not strictly speaking a mapping.

However, if $\RR$ is regarded as a mapping from $S$ to the power set of $T$, then left-totality of $\RR$ is the same as totality of this lifted function.

See the definition of a direct image mapping.

This article, or a section of it, needs explaining.In particular: "lifted function"You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Explain}}` from the code. |

### Branch

Let $A$ and $B$ be sets.

Let $f: A \to B$ be a multifunction on $A$.

Let $\family {S_i}_{i \mathop \in I}$ be a partitioning of the codomain of $f$ such that:

- $\forall i \in I: f \restriction_{A \times S_i}$ is a mapping.

Then each $f \restriction_{A \times S_i}$ is a **branch** of $f$.

## Also known as

A **multifunction** is also known as a **many-valued function**, a **multiple-valued function** or a **multi-valued function**.

On $\mathsf{Pr} \infty \mathsf{fWiki}$ the terse form **multifunction** is preferred.

When the number of values is known to be $n$, the **multifunction** can be referred to as an **$n$-valued function**.

## Examples

### Arbitrary Multifunction

Consider the implicit function:

- $y^2 = x + 2$

For $x > 2$, there are $2$ values of $y$ for every $x$.

Hence on that domain $y$ is a **two-valued function** of $x$.

## Also see

- Results about
**multifunctions**can be found**here**.

## Sources

- 1973: G. Stephenson:
*Mathematical Methods for Science Students*(2nd ed.) ... (previous) ... (next): Chapter $1$: Real Numbers and Functions of a Real Variable: $1.3$ Functions of a Real Variable: $\text {(a)}$*Many-valued Functions* - 1981: Murray R. Spiegel:
*Theory and Problems of Complex Variables*(SI ed.) ... (previous) ... (next): $2$: Functions, Limits and Continuity: Single- and Multiple-Valued Functions - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**function (map, mapping)** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**function (map, mapping)**