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 $f:\mathbb{N}\to Y$ be a function. Then the image $f(\mathbb{N})$ is at most countable.
• Let $X$ be a countable set, and let $f:X\to Y$ be a function. Then $f(X)$ is at most countable.
• Let $X$ and $Y$ be countable sets. Then $X\cup Y$ is a countable set.
• The integers $\mathbb{Z}$ are countable.
• The set $A:=\{(n,m)\in \mathbb{N}\times \mathbb{N}:0\le m\le n\}$ is countable.
• The set $\mathbb{N}\times \mathbb{N}$ is countable.
• The rationals $Q$ are countable.
• The set of all functions from $0,1$ to $\mathbb{N}$ is countable.
• The set of all functions from $\mathbb{N}$ to $0,1$ is uncountable, representing an arbitrary binary string.

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

Once we have a bijection from $\mathbb{N}$ to $\mathbb{N}\times \mathbb{N}$, we can have a bijection to the rationals, which are defined as a pairing of two numbers to make a quotient.