If $X$ and $Y$ are countable, then $X \times Y$ is countable.

Question

(Analysis 1 by Tao) Exercise 8.1.8 Use Corollary 8.1.13 to prove Corollary 8.1.14.

Corollary 8.1.13. The set $\mathbb{N} \times \mathbb{N}$ is countable.

Corollary 8.1.14. If $X$ and $Y$ are countable, then $X \times Y$ is countable.

For the proof of Corollary 8.1.13, the book shows that $\mathbb{N} \times \mathbb{N} = A \cup B$, where $A = \{(n,m) \in \mathbb{N} \times \mathbb{N} : 0 \le m \le n\}$ and $B = \{(m,n) \in \mathbb{N} \times \mathbb{N} : 0 \le n \le m\}$.

I also know that $X$ and $Y$ are countable, there exists bijections from $\mathbb{N}$ to $X$ and $\mathbb{N} $ to $Y$. To finish the proof, I need to find the bijection from $\mathbb{N} \times \mathbb{N}$ to $X \times Y$, but I do not know how to get this.

Any help would be appreciated.

Answer 1

Denote by $f:\Bbb N \to X$ , $g:\Bbb N \to Y$ the bijections from $\Bbb N$ to $X,Y$.

Then $h : \Bbb N \times \Bbb N \to X \times Y$ which is defined by $h(n,m)= (f(n),g(m))$ is the bijection you are looking for.

Answer 2

Enumerate following the systematic sequence of $(x_i,y_j)$ indexes

$$11,\ 21,12,\ 31,22,13,\ 41,32,23,14,\ 51,42,33,24,15,\ \cdots$$

or any other enumeration of $\mathbb N\times\mathbb N$.

Answer 3

Another way without the corollary but intuitive. By the fundamental theorem of arithmetic $(x,y)\mapsto 2^{x'}3^{y'}$ defines an injection from $X\times Y$ into $\Bbb N$ given bijections $x\mapsto x',y\mapsto y'$ from $X,Y$ to $\Bbb N$ respectively.