Cardinality of the set of functions that maps from $\mathbb{N}$ to $\{1,2,3\}$.

Question

I know that from the other direction, the cardinality of the set of function that maps from $\{1,2,3\}$ to $\mathbb{N}$ is $\mathbb{N}\times\mathbb{N}\times\mathbb{N}=|\mathbb{N}|$, however, I don't know how should I work from the other direction.

Answer 1

The set of functions mapping $X\to Y$ is often denoted $Y^X$. It is known that $|Y^X| = |Y|^{|X|}$.