\( \newcommand{\cat}[1] {\mathrm{#1}} \newcommand{\catobj}[1] {\operatorname{Obj}(\mathrm{#1})} \newcommand{\cathom}[1] {\operatorname{Hom}_{\cat{#1}}} \newcommand{\multiBetaReduction}[0] {\twoheadrightarrow_{\beta}} \newcommand{\betaReduction}[0] {\rightarrow_{\beta}} \newcommand{\betaEq}[0] {=_{\beta}} \newcommand{\string}[1] {\texttt{"}\mathtt{#1}\texttt{"}} \newcommand{\symbolq}[1] {\texttt{`}\mathtt{#1}\texttt{'}} \newcommand{\groupMul}[1] { \cdot_{\small{#1}}} \newcommand{\groupAdd}[1] { +_{\small{#1}}} \newcommand{\inv}[1] {#1^{-1} } \newcommand{\bm}[1] { \boldsymbol{#1} } \require{physics} \require{ams} \require{mathtools} \)

Count all the things (countability propositions)

A number of propositions related to countable sets:

• All subsets of the natural numbers are at most countable.
• Let \( Y \) be a set, and let \( f: \mathbb{N} \rightarrow Y \) be a function. Then the image \( f(\mathbb{N}) \) is [...].
• Let \( X \) be a countable set, and let \( f : X \rightarrow Y \) be a function. Then \( f(X) \) is [...].
• Let \( X \) and \( Y \) be countable sets. Then \( X \cup Y \) is [...].
• The integers \( \mathbb{Z} \) are [...].
• The set \( A := \{(n, m)\ \in \mathbb{N} \times \mathbb{N} : 0 \le m \le n\} \) is [...].
• The set \( \mathbb{N} \times \mathbb{N} \) is [...].
• The rationals \( Q \) are [...].
• The set of all functions from \( {0, 1} \) to \( \mathbb{N} \) is [...].
• The set of all functions from \( \mathbb{N} \) to \( {0, 1} \) is [...].

Maybe some of these should be split up and their proofs outlined.