What is a Set-Valued Function?
In general, a set-valued function (also called a multi-valued function) has multiple inputs for a single output. In differential calculus of set-valued maps, the mapping is defined more precisely to include set-valued function derivatives.
Set-Valued Function: Simple Example
Let’s say a consumer wants to choose a cable TV service from a variety of similarly priced and affordable options. The consumer might be ambivalent about which to choose, and it may be difficult to pinpoint why they made that decision (includes sports channels? reputation of company?) but we know they will choose one. This multi-valued input to one output is the hallmark of a set-valued function.
A correspondence (from set theory) is an example of a set-valued function. A correspondence assigns a set of points to a single point; this set of points may be from the same set, or a different set entirely (Aliprantis & Border, 2006).
Fun fact: The longest name for a set-valued function is the Knaster-Kuratowski-Mazurkiewicz map, which maps from X to E, where (Beer, 1993):
- E is a locally convex space and
- X is a nonempty subset of E.
More Technical Definitions
For the most part, when you hear the term “set-valued function” it usually means it’s a multivalued function and the two terms are often used synonymously. However, there are more technical definitions that arise in areas such as differential calculus of set-valued maps. For example, Chalco-Cano et al. (2011) define a set-valued function as follows:
“A set-valued function is a function with values in Kn or KnC [Kn is the family of all nonempty compact subsets of ℝn and KnC is the family of all A ∈ Kn such that A is a convex set], the space of all nonempty compact subsets of Rn (the space of all nonempty compact convex subsets of Rn)”
Defined in this manner, it is possible to find derivatives for set-valued functions. These relatively new developments include H-differentiability (Banks et al., 1970; Hukuhara, 1967), G-differentiability (Chalco-Cano et al., 2008) and gH-differentiability (Stefanini and Bede, 2009).
Aliprantis, C. & Border, K. (2006). Infinite Dimensional Analysis. A Hitchhiker’s Guide. Springer Berlin Heidelberg.
Banks, H et al. (1970). A differential calculus for multifunctions, Journal of Mathematical Analysis and Applications 29 (1970) 246–272.
Beer, G. (1993). Topologies on Closed and Closed Convex Sets. Springer, Netherlands.
Chalco-Cano, Y. et al. (2008). On the new solution of fuzzy differential equations, Chaos, Solitons & Fractals 38 (2008) 112–119.
Chalco-Cano, Y. et al. (2011). Generalized derivative and pi-derivative for set-valued functions q. Information Sciences 181 (2011) 2177-2188.
Hukuhara, M. (1967). Integration des applications mesurables dont la valeur est un compact convexe, Funkcialaj Ekvacioj 10. 205–223.
Stefanini, L. & Bede, B. (2009). Generalized Hukuhara differentiability of interval-valued functions and interval differential equations, Nonlinear Analysis 71 1311–1328
Walker, M. (2020). Correspondences. Retrieved October 20, 2020 from: http://www.u.arizona.edu/~mwalker/02_Equilibrium%20Existence/Correspondences.pdf
Stephanie Glen. "Set-Valued Function (Multi-Valued)" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/set-valued-function/
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