Is the set of all functions from $\mathbb{N}$ to $\{0,1\}$ countable or uncountable? [duplicate]

Hint: You can think of each function as a binary representation; then use Cantor's diagonal argument to show that it can't be countable.

Set up a table like $$\begin{align}&a_{11}a_{12}a_{13}\ldots\\ &a_{21}a_{22}a_{23}\ldots\\ &a_{31}a_{32}a_{33}\ldots \end{align}$$ where each $a_{ij}\in\{0,1\}$, and use the diagonal argument with the $a_{jj}$.

Hint: Show that the function $F\colon\{0,1\}^\mathbb N\to\mathcal P(\mathbb N)$ defined by $F(f)=\{n\in\mathbb N\mid f(n)=1\}$ is a bijection.

Use Cantor's theorem to show that $\mathcal P(\mathbb N)$ is uncountable and deduce the same on $\{0,1\}^\mathbb N$.