Past exam of the mathematics course of the University of Cambridge / 2023 / ia / Paper 4 / 8D / Solution

If $A_n$ are countable, choose enumerations and map each element of ${\bigcup }_n{A}_n$ to the first pair $(n,k)$ at which it occurs. Since ${\mathbb{N}}^2$ is countable, the union is countable.

A periodic function of period $k$ is determined by its $k$ values on a complete residue system. Thus all periodic functions form a countable union over $k$ of the countable sets ${\mathbb{N}}^k$, and are countable.

The set of bijections $\mathbb{N}\to \mathbb{N}$ is uncountable. Indeed, each binary sequence determines a bijection that swaps $2j-1$ and $2j$ exactly when its $j$th bit is one. This is an injection from the uncountable set of binary sequences.