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Sorry if it seems a silly question.

I have a function $f$ that returns a certain set of elements that are all in the same domain. The exact number of element depends on the input. For example $f(x) = \{0,1\}$ for $x = 42$ and $f(x) = \{0,1,2\}$ for $x = 3.14$. On this example, how do I write the domain of the output ?

I would expect something like $f:\mathbb{R} \longrightarrow \mathbb{N}^k$ or $f:\mathbb{R} \longrightarrow k^\mathbb{N}$ with $k \in \mathbb{N}$, but I thinks these notations are wrong... What is the right one ?

Welgriv
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1 Answers1

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Note that $\mathbf N^k$ is the set of $n$-tuples (not $n$-subsets) of $\mathbf N$, if you want subsets, the usual notation for the so called powerset, the set of all subsets of $\mathbf N$ is $\mathscr P(\mathbf N)$. If you want restrict the range to finite subsets of $\mathbf N$, something like $\mathscr P_{\mathrm{fin}}(\mathbf N)$ or $\mathscr P_{<\aleph_0}(\mathbf N)$ might be understood. But as the is no general notation for that, before using something like that, I'd define it for the reader. That is write something like:

Let $\mathscr P_{\textrm{fin}}(\mathbf N) = \{A \subseteq \mathbf N : A \text{ is finite}\}$ denote the set of finite subsets of $\mathbf N$. Consider a function $f \colon \mathbf R \to \mathscr P_{\textrm{fin}}(\mathbf N)$ ...

martini
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  • So in my example the correct notation would simply be $f: \mathbb{R} \longrightarrow \mathcal{P}(\mathbb{N})$ ? Or $f: \mathbb{R} \longrightarrow \mathcal{P}_{fin}(\mathbb{N})$ more specifically ? – Welgriv Feb 16 '22 at 16:34
  • It depends if you want to allow for infinite sets. And, as I said above, in case of the latter, I'd define it in the text. – martini Feb 16 '22 at 16:57