# What is a set function that returns another set of points called?

I have a set of points $S = \{x_i\}_{i = 1}^m, x_i \in \mathbb{R}^n \forall i$. Now, I have a set function $f$ which operates as follows: $$f(S) = GX^T$$ where $G \in \{0,1\}^{m\times m}$ and $X = [x_1 x_2 \ldots x_m]$.

Is there a specific name for such $f$'s?

• Set-valued function? Jul 22, 2013 at 7:51
• The set-valued functions that I have come across don't seem to have set-valued input to the functions. I mean, the kind of set-valued functions studied seem to be $f:X\rightarrow 2^Y$ but not $f:2^X\rightarrow 2^Y$. Jul 22, 2013 at 8:05

The function $f$ does not return a set of points: it returns an $m\times 1$ matrix of real $n$-tuples. And the input is not a set $S=\{x_k:k=1,\dots,m\}\subseteq\Bbb R^n$: you’re using the indexing of the $x_k$ in an essential way, so the input to $f$ is actually an $m$-tuple of vectors in $\Bbb R^n$. Specifically,
$$f:\left(\Bbb R^n\right)^m\to\left(\Bbb R^n\right)^{m\times 1}:\langle x_1,\dots,x_m\rangle\mapsto G\begin{bmatrix}x_1\\x_2\\\vdots\\x_m\end{bmatrix}\;.$$