cardinality of sets intersections

Question

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 $B$?