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.

(In the following, countable means what is often referred to as countably infinite, i.e. countable and not finite)
A set $A$ is called countable if there exists a bijection $f$ between $A$ and the set of natural numbers $\mathbb{N}$. In other words, $A$ is countable if there is some mapping $f \,:\, A \to \mathbb{N}$ such that $f$ hits every element of $\mathbb{N}$ exactly once. Note that any such $f$ has a uniquely defined inverse $f^{-1}$.
It follows that there exists a bijection $f_{A,B}$ between any two countable sets $A$ and $B$, because if $f_A$ is a bijection from $A$ to $\mathbb{N}$, and $f_b$ a bijection from $B$ to $\mathbb{N}$, then $x \mapsto f_B^{-1}(f_A(x))$ is a bijection from $A$ to $B$.
Therefore, to show that the union of two arbitrary disjoint countable sets is countable, it suffices to show that the union of two specific disjoint countable sets $A,B$ is countable. This works because if $X,Y$ are disjoint and countable, by the above there are bijections $f_X \,:\, X \to A$, $f_Y \,:\, Y \to B$, and a bijection $g \,:\, A \cup B \to \mathbb{N}$. We can therefore define a biection from $f \,:\, X \cup Y$ to $\mathbb{N}$ by setting $$
  f(z) = \begin{cases}
    g(f_X(z)) &\text{if $z \in X$,} \\
    g(f_Y(z)) &\text{if $z \in Y$.}
  \end{cases}
$$
(I leave the proof that this actually is a bijection to you)
Since we can view $\mathbb{Z}$ as the disjoint union of $\mathbb{N} = \{0,1,\ldots\}$ and $\mathbb{Z} \setminus \mathbb{N} = \{\ldots,-2,-1\}$, it therefore suffices to 


*

*show that $\mathbb{Z} \setminus \mathbb{N}$ is countable, i.e. to find a bijection between $\{\ldots,-2,-1\}$ and $\{0,1,\ldots\}$, and

*show that $\mathbb{Z}$ is countable.
