Family set of subsets of $\Bbb N$
Solution 1:
Define $C_n=g(\{n\}\times\Bbb N)$. Since $g$ is injective and$$n\ne m\implies(\{n\}\times\Bbb N)\cap(\{m\}\times\Bbb N)=\emptyset,$$the $C_n$'s are disjoint. Since $g$ is surjective, $\bigcup_{n\in\Bbb N}C_n=\Bbb N$. And, since $\{n\}\times\Bbb N$ is infinite, each $C_n$ is infinite.