Past exam of the mathematics course of the University of Cambridge / 2003 / ia / Paper 4 / 5C (Numbers and Sets)

Use the inclusive convention: a set is countable if it admits an injective function into $\mathbb{N}=\{1,2,\dots \}$. This includes every finite set and the empty set. If $j:X\to \mathbb{N}$ is injective and $Y\subseteq X$, its restriction to $Y$ is still injective, proving directly that every subset of a countable set is countable.

Enumerate pairs by diagonals. The positive-integer Cantor pairing functionis a bijection ${\mathbb{N}}^2\to \mathbb{N}$: on the diagonal $a+b-2=d$, its values are $d(d+1)/2+1$ through $(d+1)(d+2)/2$, each once, and these consecutive blocks partition $\mathbb{N}$. This proves countability without assuming any product theorem.

For a sequence of countable sets $X_1,X_2,\dots$, choose injections $j_i:X_i\to \mathbb{N}$. For $x$ in their union, let $i(x)$ be the least index containing $x$. The mapis injective: its encoded index and the injective within-set code recover $x$. Thus a countable union of countable sets is countable. Selecting the injections for an arbitrary given family uses the usual axiom of countable choice; when the encodings are supplied, the displayed construction is entirely explicit.

For finite Cartesian powers, start with the identity injection on $\mathbb{N}$ and recursively encode $(a_1,\dots ,a_n)$ as $\pi (j_{n-1}(a_1,\dots ,a_{n-1}),a_n)$. This proves that ${\mathbb{N}}^n$ is countable for every positive integer $n$ and is the constructive proof of the finite Cartesian power of a countable set property.

For each positive integer $m$, let $P_m$ consist of the functions for which $m$ is a period. Restriction to the residues $0,1,\dots ,m-1$ is a bijection $P_m\to {\mathbb{N}}^m$: any tuple extends uniquely to all of $\mathbb{Z}$ by taking residues modulo $m$, including negative arguments. Hence $P_m$ is countable. Every periodic function belongs to some $P_m$, soThere is no need to assume the chosen period is minimal; multiple descriptions only create overlap, which does not invalidate the union argument. This is an instance of the cardinality of integer-periodic function spaces principle.