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The inputs of function $f$ are a set $A$ and one natural number $n$.

Plus, its outputs are $n$ 3-dimensional coordinates.

For example, $A$ is defined by ~~~. Then, $f(A, 4) = \{(x_1, y_1, z_1), \ldots, (x_4, y_4, z_4)\}.$

In this case, is it correct to write like the following? $$f:(A, \mathbb{N}) \to \mathbb{R}^{3\times\mathbb{N}}$$

Danny_Kim
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  • Keep in mind that $\mathbb{R}^{3\mathbb{N}}\neq\mathbb{R}^{3n}$ and that $\mathbb{R}^{3n}$ is a coordinate with $3n$ components, not $n$ components of 3. – 高田航 Oct 29 '17 at 14:56
  • In my opinion more standard notation would be $f:\text{Set}\times \mathbb{N} \to \mathcal{P}(\mathbb{R}^3)$ where $\mathcal{P}(\mathbb{R}^3)$ is the powerset of real space, and $\text{Set}$ is class of all sets (interpret it as you find appropriate). – user340297 Oct 29 '17 at 14:59
  • Careful, there's a difference between $(A, \mathbb N)$ and $A\times\mathbb N$ – Bart Michels Oct 29 '17 at 15:04
  • @user340297 Powerset of $\mathbb{R}^3={x,y,z}$ is ${\emptyset, {x}, {y}, {z}, {x,y}, {y,z}, {x,z}, {x,y,z}}$, isn't it? – Danny_Kim Oct 29 '17 at 15:55
  • @barto Thank you, from now on, I will use $\times$ operator instead of tuple. – Danny_Kim Oct 29 '17 at 16:02
  • @高田航 If so, how can I write $\mathbb{R}^{3\times\mathbb{N}}$ correctly? – Danny_Kim Oct 29 '17 at 16:04

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No. I think it's hard to write a "formal definition" here.

Let $\mathcal{A}$ be the set of sets $A$ for which the definition makes sense. Then the domain of your function $f$ is $ \mathcal{A} \times \mathbb{N}$, the set of ordered pairs $(A,n)$. For each particular pair the codomain is $\mathbb{R}^n$ so you could write the codomain of $f$ as the union of those spaces, but that probably wouldn't be helpful.

I would suggest writing about a sequence of functions $f_n: \mathcal{A} \rightarrow \mathbb{R}^n$, one for each $n$.

Ethan Bolker
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  • This is good enough for me, as it stands right now. – Lubin Oct 29 '17 at 15:27
  • @Lubin OK if you say so. – Ethan Bolker Oct 29 '17 at 17:19
  • @EthanBolker I am sorry for not understanding of using $f_n$. Is it fine if I define $$f = \bigcap_n f_n?$$ – Danny_Kim Oct 29 '17 at 23:28
  • @Danny_Kim I think you can write $f$ as the union (not the intersection) of the $f_n$ but I continue to wonder why you would want to. You really have a set of functions, one for each $n$. I don't see what you gain by trying to turn that set of functions into a single function. – Ethan Bolker Oct 29 '17 at 23:34
  • @EthanBolker Okay, thank you for fast responding. Can I ask one more thing? Is there any word like powerset admitting repetition of elements? For example, if $A={x}$, then the power set of $A$ is ${\emptyset, {x}}$. However, I want to know the word that represents ${\emptyset, {x}, {x, x}, {x,x,x} \ldots }$. – Danny_Kim Oct 29 '17 at 23:38
  • Look at "multiset", or sometimes "bag": https://en.wikipedia.org/wiki/Multiset. You seem to have invented something you could call a power-multiset. – Ethan Bolker Oct 29 '17 at 23:42