I'm thinking about the proof of the following theorem:
If $\mathcal A$ is a denumerable family of denumerable sets then $\bigcup \mathcal A$ is denumerable. (denumerable means that there is a bijection to $\mathbb N$)
The proof shows $|\mathbb N| \leq |\bigcup \mathcal A|$ and $|\mathbb N| \geq |\bigcup \mathcal A|$ rather than giving an explicit bijection $f: \mathbb N \to \bigcup \mathcal A$.
Question 1: In this case, is it possible to give an explicit bijection?
Question 2: In general, is it possible to find a bijection between set $A$ and $B$ if we know that $|A| = |B|$?