Assume we need to prove the countability of $(\Bbb{N}\times\Bbb{N}) \cup \Bbb{N}$
As far as I know, $\Bbb{N}\times\Bbb{N}$ will result in pairs such as $\langle\,1,1\,\rangle$ for all possible $\Bbb{N}$.
The resulting set of the union, would then contains all possible $\Bbb{N}$ values plus all the pairs.
To prove the countability of the sets, I usually draw in extension mode a bijective function, and then I can assume that it is countable as the cardinality are the same.
The problem here is that I have no idea how to map to a set which has pairs and singleton - is that even possible ? Any hint ?
Guide:
Prove that $\mathbb{N} \times \mathbb{N}$ is countable.
Prove that union of countable set is countable.
Remark: $\mathbb{N} \times \mathbb{N}$ is known as the cartesian product rather than the power set.