Countability (show set is countable) Show that the set $\mathbb{Z}_+\times\mathbb{Z}_+$ is countable.?
To solve this you have to show a one to one correspondence.
$\mathbb{Z}_+\times\mathbb{Z}_+\to\mathbb{Z}_+$ Then my book recommends using $f(m,n) = 2^m\times3^n$ (or any other primes) to show it is one to one. 
Where do these numbers come from, how do you know to use primes? Where is the thinking behind this? 

It isn't important that you use the primes 2 or 3. The fact that there is a one to one correspondence between $\mathbb{Z}_+ \times \mathbb{Z}_+$ and $f(\mathbb{Z}_+,\mathbb{Z}_+)$ is a consequence of the unique factorization in $\mathbb{Z}$. The map
$$f: \mathbb{Z}_+ \times \mathbb{Z}_+ \to f(\mathbb{Z}_+,\mathbb{Z}_+) $$
is clearly surjective. It is also injective because $2^{n_1}3^{m_1}$ can only be equal to $2^{n_2}3^{m_2}$ when $n_1=n_2$ and $m_1=m_2$ because of unique factorization. Therefor there is a bijective correspondence between $\mathbb{Z}_+ \times \mathbb{Z}_+$ and $f(\mathbb{Z}_+,\mathbb{Z}_+)$. That last one is countable as a subset of the countable set $\mathbb{Z}_+$ and therefor also $\mathbb{Z}_+ \times \mathbb{Z}_+$ is countable.
