Is pair $(m, -m)$ which $m\in \mathbb{Z}$ is isomorphic to $\mathbb{Z}$? Is $\{(m, -m)\mid m\in \mathbb{Z}\}$ is isomorphic to $\mathbb{Z}$?
What i know about isomorphic is, simply it is a bijective homomorphism. 
How one can define isomorphism map $\phi$ in above case?

Yes it is ! Just take $$\varphi: \mathbb Z\longrightarrow M$$
defined by $$\varphi(m)= (m,-m).$$
The surjectivity is obvious. For the injectivity, it's easy to prove that $\ker \varphi=\{0\}.$
