Math and science::Analysis::Tao::08. Infinite sets

# 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.