Finite Cartesian Product of Countable sets is countable?

Is the finite Cartesian Product of Countable sets countable?

If possible, could you give a bijection that would work for any example?

Any help would be appreciated.

Solution 1:

Hint.

The answer is positive.

$f_2$ defined from $\mathbb N \times \mathbb N $ to $\mathbb N$ by $$ f_2(p,q)= 2^{p-1}(2q-1)$$ is a bijection.

The case of $n$ countable sets can then be considered by induction. For example for $n=3$ you can define $$f_3(p,q,r)=f_2(f_2(p,q),r)$$ which is a bijection from $\mathbb N^3$ to $\mathbb N$.

More generally $$f_{n+1}(p_1, \dots ,p_{n+1})= f_2(f_{n}(p_1, \dots ,p_{n}),p_{n+1})$$