how to prove that $\#\mathcal{P}(\mathbb{N}) = \#\mathbb{R}$? I'm thinking in associate a subset of $\mathbb{N}$ in a real number by the decimal representation, but i'm failed.
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Try binary representation – user126154 Apr 18 '20 at 16:33
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5Does this answer your question? The set of real numbers and power set of the natural numbers – Physical Mathematics Apr 18 '20 at 18:38
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Here's an injective map $\mathcal P(\Bbb N)\to\Bbb R$, showing $\#\mathcal P(\Bbb N)\le\#\Bbb R$:
$$S\mapsto \sum_{k\in S}3^{-k} $$
Here's an injective map $\Bbb R\to \mathcal P(\Bbb Q)$, showing $\#\Bbb R\le\#\mathcal P(\Bbb Q)$:
$$\alpha\mapsto \{\,x\in \Bbb Q\mid \alpha<x\,\}.$$
As $\#\Bbb N=\#\Bbb Q$, the result follows.
J.G.
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Hagen von Eitzen
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