Why is $\{f\,|\,f\colon A\to\mathbb N\}$ not uncountable?

Question

Let $S=\{\,f\,|\,f\colon A\to\mathbb N\}$, where $A=\{1,2\}$.

I thought cardinality of $S$ is $2^{|\mathbb N|}=\aleph_0$. But my friend told that my answer was wrong.

Please help me where is I am wrong.

Answer 1

The cardinality of the set of functions from a set $A$ to a set $B$ is $|B|^{|A|}$.

In particular in your case the size of $S$ is $|\mathbb{N}|^2$ (not $2^{|\mathbb{N}|}$) which is the size of $\mathbb{N}\times\mathbb{N}$ (hence countable).