Possible Duplicate:
countably infinite union of countably infinite sets is countable
How to prove the union of two disjoint denumerable sets is denumerable?
I want a direct proof
Possible Duplicate:
countably infinite union of countably infinite sets is countable
How to prove the union of two disjoint denumerable sets is denumerable?
I want a direct proof
By hypothesis you have disjoint sets $A=\{a_n:n\in\Bbb N\}$ and $B=\{b_n:n\in\Bbb N\}$. You want a bijection $f:\Bbb N\to A\cup B$.
HINT: There is an easy way to define $f$ so that it maps the set of even integers in $\Bbb N$ bijectively to $A$ and the odd ones bijectively to $B$. Where do you think that $f$ should send $2n$? What about $2n+1$?
Added: There are at least a couple of ways to write down such a function. You can say:
For each $n\in\Bbb N$ let $f(2n)=\text{thing}_1$ and $f(2n+1)=\text{thing}_2$.
Alternatively, you can give a two-case definition of $f$ like this:
Let $f:\Bbb N\to A\cup B:n\mapsto\begin{cases}\text{formula}_1,&\text{if }n\text{ is even}\\\text{formula}_2,&\text{if }n\text{ is odd}\;.\end{cases}$
The formulas might involve division by $2$, among other things.