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I was trying to prove that $P \left( \mathbb{N} \right)$ is bijective with the interval $\left[ 0, 1 \right)$. I wish to employ Schoder-Berstein theorem which states that if there there are two one-to-one functions between two sets, then those sets are bijective.

To obtain the first one-to-one function, I formed $f: P \left( \mathbb{N} \right) \rightarrow \left[ 0, 1 \right)$ as

$$f \left( S \right) = \begin{cases} 0, & S = \emptyset \\ 0.a_1a_2a_3\cdots, & S \neq \emptyset \end{cases}$$

where $a_n = \begin{cases} 1, & n \in S \\ 0, & n \notin S \end{cases}$.

Clearly, this is a one-to-one function, as commented upon in Bijection from $\mathcal{P}(\mathbb{N})$ to $(0,1)$.

However, I find it difficult to create the other one-to-one function. I tried to form $g: \left[ 0, 1 \right) \rightarrow P \left( \mathbb{N} \right)$ as

$$g \left( x \right) = \begin{cases} \emptyset, & x = 0 \\ \left\lbrace a_1, a_2, \cdots \right\rbrace, & x = 0.a_1a_2\cdots \end{cases}$$

However, clearly, this is not one-to-one since $0.1$ and $0.11$ are mapped to the same set $\left\lbrace 1 \right\rbrace$. Also, it is not well-defined since $0.01$ is not mapped anywhere.

Any hints for constructing this one-to-one function will be appreciated.

Note: I would like to use the decimal expansion only for the construction, and not base-two expansion.

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Consider the base-two expansion of $x\in [0,1)$: $$x=0.b_1b_2\cdots=\sum_{n=1}^\infty b_n2^{-n}$$ with each $b_n\in\{0,1\}$. Consider the set $\{n\in\Bbb N:b_n=1\}$.

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An injective map $g: \left[ 0, 1 \right) \rightarrow P \left( \mathbb{N} \right)$ is $$g(x)=\{p_n^{\lfloor 10^n x \rfloor}: n\in\mathbb{N}\}$$ where $p_n$ is the $n$-th prime number and with the restriction that we forbid decimal representations with a trailing infinite sequence of $9$s.

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  • $\begingroup$ Consider $x = 0.0999999\cdots = 0.1$. Then, $g \left( x \right) = \left\lbrace 2 \right\rbrace$ by considering the decimal expansion $0.1$. On the other hand, if we consider the other decimal expansion, we will have $g \left( x \right) = \left\lbrace 1, 3^9, 5^{99}, 7^{999}, \cdots \right\rbrace$, which are clearly two different sets. So, $g$ is not well-defined. $\endgroup$ – Aniruddha Deshmukh Oct 28 '19 at 5:13
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    $\begingroup$ @AniruddhaDeshmukh Please se my edit. Conventionally, the decimal representation without trailing 9's is preferred. See en.wikipedia.org/wiki/… $\endgroup$ – Robert Z Oct 28 '19 at 6:31

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