[Hi everyone, I am not sure of how to apply question 2, and Proposition 9.2.4(2) [which means Any subset A of a countable set B is countable] shown in 1 of the pictures above to solve questions 3(a) and (b)..
For 3(a) I believe I need to apply the fact that |A x C| = |B x D| as part of my proof... but how do I even do that? The even bewildering/confusing part is how do I even apply 3(a) and Proposition 9.2.4(2) shown above in one of the pictures for 3(b)?
Will appreciate it a lot if anyone can help me out with this. Thanks a lot!]
Image Links of the question and proposition: Question: https://i.stack.imgur.com/Dxtmg.jpg Proposition: https://i.stack.imgur.com/89P2M.jpg
You can start by showing that $\mathbb{N}\times\mathbb{N}$ is countable. This can be done by finding an injective function from $\mathbb{N}$ to $\mathbb{N}\times\mathbb{N}$ and another injective function from $\mathbb{N}\times\mathbb{N}$ to $\mathbb{N}$, and then applying the Cantor-Schröder-Bernstein theorem to show that there exists a bijection $f\colon \mathbb{N}\to\mathbb{N}\times\mathbb{N}$.
Then, since $A$ and $B$ have the same cardinality as $\mathbb{N}$, from Question 2 we can conclude that $A\times B$ has the same cardinality as $\mathbb{N}\times\mathbb{N}$, so it is countable.
Finally, for question 3(b), let $A_0$ and $B_0$ be countable sets. Let $A$ and $B$ be countably infinite sets such that $A_0\subseteq A$ and $B_0\subseteq B$. From question 2, we know that $A\times B$ is countable and since $A_0\times B_0 \subseteq A\times B$, Proposition 9.2.4(2) allows us to conclude that $A_0\times B_0$ is countable.