Q:

Prove: if $f:\mathbb{N} \rightarrow A, g:\mathbb{N} \rightarrow B$ are surjections, there exists a surjection $h:\mathbb{N} \rightarrow A \cup B$

I chose my own sets here for A and B as countably infinite pairwise disjoint subsets of $\mathbb{N}$. Can I do this with finite subsets and get an easier answer with the same result? 

Suppose $A = \mathbb{N}_e$ such that $\mathbb{N}_e$ is the set of all even natural numbers.  $\mathbb{N}_e \subset \mathbb{N}$
Also suppose $B = \mathbb{N}_o$ such that $\mathbb{N}_o$ is the set of all odd natural numbers.  $\mathbb{N}_o \subset\mathbb{N}$
$\mathbb{N}_e \cup \mathbb{N}_o = \mathbb{N}$
$h: \mathbb{N} \rightarrow A \cup B$ is given by
$h(x) = $
$f(\frac{x}{2})$ if $x\in \mathbb{N}_e$
$g(\frac{x+1}{2})$ if $x\in \mathbb{N}_o$
Let $n \in \mathbb{N}$. If $n \in \mathbb{N}_e$, consider $x = 2n$
$f(x) = \frac{2n}{2} = n$
Let $n \in \mathbb{N}$. If $n \in \mathbb{N}_o$, consider $x = 2n-1$
$g(x) = \frac{(2n-1)+1}{2} = n$
So $h: \mathbb{N} \rightarrow A \cup B$ is surjective. 

I'm kind of iffy on notation here.  I'm wondering when it came to constructing the $h$ function, why it wasn't $f(x)$ if $x \in A$ or something like that.  I get these two different things confused all of the time. 

UPDATE
I understand what's wrong with this.  I can't decide what $A$ and $B$ are because they might not actually even by numbers.  The function $h$ assigns them numbers to cover everything.  Kind of like a room of boys and girls and $f$ maps to all of the boys, and $g$ maps to all the girls?  $h$ will map to the union, which is the entire room hence it is surjective.  

A:

HINT: You are supposed to be proving a general theorem about all pairs of sets $A$ and $B$ such that there are surjections $f:\Bbb N\to A$ and $g:\Bbb N\to B$, so you are not free to choose specific sets $A$ and $B$. Thus, your argument gets off very much on the wrong foot. 
It is useful, however, to look at the sets $\Bbb N_e=\{2n:n\in\Bbb N\}$ and $\Bbb N_o=\{2n+1:n\in\Bbb N\}$. Note that there are bijections 
$$\varphi:\Bbb N_e\to\Bbb N:n\mapsto\frac{n}2$$
and
$$\psi:\Bbb N_o\to\Bbb N:n\mapsto\frac{2n-1}2\;.$$

Verify that $\varphi$ and $\psi$ are bijections.   
Show that $f\circ\varphi$ is a surjection from $\Bbb N_e$ to $A$.   
Show that $g\circ\psi$ is a surjection from ... ?   
Use the previous two points to construct a surjection from $\Bbb N$ to $A\cup B$.

