
Realtion  Find function
I have been given a task to find a function $\displaystyle f$ which satisfies:
1. Definition area for $\displaystyle f$ is $\displaystyle \mathbb{J}$
2. Value area for $\displaystyle f$ is the amount/quantity by the equivalentclass of ~
3. f is injective
4. f is surjectiv
5. $\displaystyle f(a+b) = f(a)+f(b)$ for all $\displaystyle a$ og $\displaystyle b$ in $\displaystyle \mathbb{J}$
What is done is actually to complete the quantity theoretical constuction of all integers from the natural numbers.
I actually have no idea where to start! Feel free to point me in a direction, all help is gratefully accepted!

Could this be somewhat of a start:
$\displaystyle f: \mathbb{J} \rightarrow \mathbb{J}$
$\displaystyle j \rightarrow f(j)$
$\displaystyle f(j) = j1+2j \quad \forall j \ge 2$