Prove that the union of two disjoint countable sets is countable This is a question from my proofs course review list that I have had trouble understanding. 
I understand the concept of disjoint sets.  I'm not sure what they mean by countable. How would one prove a set is countable and furthermore, that the union of two countable sets is countable.
Please help me get started with the question by understanding this concept.

A formal proof would be something like this. If $A$ and $B$ are countable (and disjoint) sets, then there exist injective functions $f$ and $g$ from $A$ and $B$, respectively, to $\mathbb{N}$. Let $C=A\cup B$ and define a function $h:C\rightarrow\mathbb{N}$ as follows:
$$h(z) = \begin{cases}2f(z) & \text{if } z\in A\\ 2g(z)+1 & \text{if } z\in B\end{cases}$$
Then, if $x\neq y$, we clearly have $h(x)\neq h(y)$ since if $x,y\in A$ this follows from the fact that $f$ is injective (analogously for $x,y\in B$) and if $x\in A$ and $y\in B$, we have that $h(x)$ is even while $h(y)$ is odd (analogously for $x\in B$, $y\in A$). Hence $h$ is an injective function from $C$ to $\mathbb{N}$ and thus, by definition, $C$ is countable.
