Show that the set of all subsets of $\mathbb{N}$ is uncountable, and that the set of all finite subsets of $\mathbb{N}$ is countable.

Let $X$ be the set of all bijections from $\mathbb{N}$ to $\mathbb{N}$, and let $Y \subset X$ be the set

$$Y=\{f \in X \mid \text { for all but finitely many } n \in \mathbb{N}, f(n)=n\}$$

Show that $X$ is uncountable, but that $Y$ is countable.