Definition:Left-Total Relation/Multifunction

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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

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