# 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.

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### 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 (multi)function** of $x$.

### Unit Circle

Consider the equation for the unit circle whose center is at the origin of the Cartesian plane:

- $x^2 + y^2 = 1$

This can be considered as the graph of a **multifunction** whose domain is the closed real interval $\closedint {-1} 1$ and whose branches are:

\(\ds y\) | \(=\) | \(\ds +\sqrt {1 - x^2}\) | ||||||||||||

\(\ds y\) | \(=\) | \(\ds -\sqrt {1 - x^2}\) |

## 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)** - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**multiple-valued function (many-valued function)** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**function (map, mapping)** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**multiple-valued function (many-valued function)**