Originally Posted by

**hatsoff** This is quite inappropriate. You should *never* demand a full proof. However, since I feel your pain about the exam, I will say this much:

Since $\displaystyle \mathbb{N},\mathbb{Z}$ are equinumerous, we may let $\displaystyle \varphi:\mathbb{N}\to\mathbb{Z}$ be any bijection, and define the binary operation $\displaystyle \diamond:\mathbb{Z}\times\mathbb{Z}\to\mathbb{Z}$ by $\displaystyle \varphi(m)\diamond\varphi(n)=\varphi(m+n)$ for all $\displaystyle m,n\in\mathbb{N}$. Then it follows immediately that $\displaystyle \varphi$ is an isomorphism. It remains to prove that $\displaystyle \diamond$ is well-defined.