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I've learned over the course of the last years that some mapping $\lambda$ denoted :

$$ \lambda : \mathbb{N} \rightarrow 2^{\mathbb{N}}.$$

essentially means that for every natural number, you assign (or map) some set of natural numbers to it. At multiple times, I've wondered if there's some meaning behind this notation (more precisely the $2^{\mathbb{N}}$ part) that I'm not understanding, as it does not seem as straightforward as say :

$$ f : n \rightarrow 2n$$

(also just noted as $f(n) = 2n$), where you clearly map any $n$ to $2n$. Is this $2^{\mathbb{N}}$ just some notation that one has to get used to, or does it make sense somehow to note it this way? If it does make sense, why the choice of exponentiation, and why the $2$?

Xander Henderson
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J. Schmidt
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    In your examples, the $2^\mathbb{N}$ and $2n$ are not used in the same way. One is the codomain that $\lambda$ maps to, and one is the specific value that $f$ maps its argument to. As Dave explains in their answer, the specification of $\lambda$ is strange, because although it uses $2^\mathbb{N}$ as its codomain, it does not give a domain; it gives a specific value $n$. – Brian Tung May 29 '20 at 17:55
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    Compare:https://math.stackexchange.com/q/901735/87023 https://math.stackexchange.com/q/1442509/87023 https://math.stackexchange.com/q/104522/87023 https://math.stackexchange.com/q/2569341/87023 – Chris Culter May 29 '20 at 18:00
  • @ChrisCulter nice ! You can note this question as a duplicate I suppose, I'll also add these in my accepted answer. – J. Schmidt May 29 '20 at 18:08
  • This one, too: https://math.stackexchange.com/questions/104522/ – Xander Henderson May 29 '20 at 18:26
  • And this one: https://math.stackexchange.com/q/901735/468350 – Xander Henderson May 29 '20 at 18:27

1 Answers1

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$2^{\mathbb N}$ denotes the power set of $\mathbb N$, i.e. the set of all subsets of $\mathbb N$. This notation is used because, for a finite set $A$, the size of the power set $2^A$ of $A$ is $2^{|A|}$.

However, $\lambda:n\to 2^{\mathbb N}$ still seems like weird notation to me: the $n$ is an element of $\mathbb N$ (presumably) but $2^{\mathbb N}$ is a set. Should it maybe be $\lambda:\mathbb N\to 2^{\mathbb N}$?

Dave
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  • Yes, so the original equation was $\lambda : E \rightarrow 2^{\mathbb{N}}$, where $E$ denotes the edge set of a graph. I tried to generalize my question by putting just some $n$ instead, but I forgot (as you and Brian Tung noted) that it should be a set still, instead of a specific value (I'll edit this back in with $\mathbb{N}$). – J. Schmidt May 29 '20 at 18:01
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    $\lambda:E\to 2^{\mathbb N}$ is fine; it doesn't have to be $\mathbb N$ as the domain, it just needs to be some (nonempty) set. – Dave May 29 '20 at 18:02
  • As much as it is uncomfortable, $\lambda\colon n \to 2^{\Bbb N}$ still compiles, if one thinks that one set-theoretic of $n$ is ${0,1,\ldots, n-1}$. – Ivo Terek May 29 '20 at 19:32