Given that:

$$| A \cap \mathbb{R}| = |\mathbb{R}| $$ $$| B \cap \mathbb{R}| = |\mathbb{N}| $$

Can we say that: $$|A| = |\mathbb{R}|$$ $$|B| = |\mathbb{N}|$$ ?

If not, can we say something for sure about the cardinality of either $A$ or $B$?

  • 2
    $\begingroup$ What do you think? What about $B=\mathbb{Z}$ or $B=\mathbb{R}?$ $\endgroup$ – mfl Jan 27 at 22:26
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    $\begingroup$ What about $A := \mathcal{P}(\mathbb{R}) \cup \mathbb{R}$, then $|A| \geq |\mathbb{R}|$ $\endgroup$ – Nathanael Skrepek Jan 27 at 22:28
  • $\begingroup$ Well, when you put it this way...It seems obvious now, thank you guys, you can post it as an answer $\endgroup$ – HeyJude Jan 27 at 22:49
  • $\begingroup$ What if $B = \mathbb N \cup (\mathbb C \setminus \mathbb R)$. $\endgroup$ – fleablood Jan 27 at 23:01

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