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We suppose that we have a set $\!S \subseteq \mathcal{P}(\mathbb{R})$, so that the following properties are verified:

1). $\space \!S \sim \mathbb{R}$

2). if $\space \!X,\!Y \in \!S \space and \space \!X \neq \!Y, \space$ then $\space \!X \cap\!Y= \varnothing$

3). if $\!X \in \!S$, then $\space \!X \sim \mathbb{R}$

I don't event know where to begin

Dknot
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1 Answers1

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The cardinal of $\mathbb{R}-\mathbb{Q}$ is the cardinal of $\mathbb{R}$, for every $x\in \mathbb{R}-\mathbb{Q}$, consider $S_x=\{x+i,i\in\mathbb{N}\}$ and $S=\cup_xP(S_x)$. The cardinal of $P(S_x)$ is the cardinal of $P(\mathbb{N})$ is the cardinal of $\mathbb{R}$.

  • Not completely sure if this set fits 3rd point. Because lets say we take $!S_{x}$ as out element. $!S_{x} \in S$ but $!S_{x} \not\sim \mathbb{R}$ – Dknot Mar 22 '20 at 09:28