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)