What does it mean to prove that the addition of two countable sets is countable? How Should I prove that $\mathbb{Q} + i\mathbb{Q}$ is a countable set? I've already proven that $\mathbb{Q}$ is countable.
 A: If you know that $\Bbb{Q}$ is countable, it follows that $i\Bbb{Q}$ is countable as well (since $i\Bbb{Q} = \{ iq : q \in \Bbb{Q}\}$, whereby $\Bbb{Q}$ and $i\Bbb{Q}$ have the same cardinality).
And if you know that these two sets are countable, then by definition there exists a bijection between $\Bbb{N}$ and $\Bbb{Q}$, and there exists a bijection between $\Bbb{N}$ and $i\Bbb{Q}$.
And you know that $\Bbb{Z}$ is countable and in Bijection with $\Bbb{N}$ as well, so you can set up a bijection between $\Bbb{Z}$ and $\Bbb{Q}$ $\cup$ $i\Bbb{Q}$ easily, like this:
Take $x \in \Bbb{Q}$ $\cup$ $i\Bbb{Q}$. If $x \in \Bbb{Q}$, then map $x$ to its counterpart in $\Bbb{N}$ according to the bijection you know that exists. If $x \in i\Bbb{Q}$, then map $x$ to its counterpart in $\Bbb{N}$ according to the other bijection you know exists. Let that counterpart be $v$. Map $v \in \Bbb{N}$ to $-v \in \Bbb{Z}$.
Now you're mapping the $\Bbb{Q}$ part of $\Bbb{Q}$ $\cup$ $i\Bbb{Q}$ to the positive part of $\Bbb{Z}$, and you're mapping the $i\Bbb{Q}$ part to the negative part of $\Bbb{Z}$. And you're done.
You can make this proof a lot more elegant or short, but I tried to be clear. You'll also have to deal with the case of which element in  $\Bbb{Q}$ $\cup$ $i\Bbb{Q}$ maps to $0$ (this is easy -- you'll just have to declare what it maps to, and adjust your bijections slightly, like by shifting one map over by $1$ -- e.g. if $q \in \Bbb{Q}$ mapped to $n$, map it to $n+1$ or something like that).
