# cardinality of sets intersections

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 $$B$$?

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