Writing each integer as a rational number

Question

Suppose the rational number is defined as $\mathbb{Q}:=X / \sim$ where $X:=\mathbb{Z} \times(\mathbb{Z} \backslash\{0\})$ and $\sim$ is the equivalence relation on X and assume all properties of addition and multiplication of the integers is defined, how do I show that for every $n \in \mathbb{Z}$, there is a rational number $[(a, b)] \in \mathbb{Q}$ that corresponds to $n$ rigorously? I know that the answer is $[(n, 1)]$ but what additional things I have to show in order to answer the question? I know I might have to show that this $[(n, 1)]$ is unique but I cannot think of anything else. Can someone hint at it to me?

Answer 1

All you need to show is that the mapping $\ \varphi:\mathbb{Z}\rightarrow X/\sim\ $ defined by $$ \varphi(n)=[(n,1)]=\left\{(mn,m)\,|\,m\in\mathbb{Z}\setminus\{0\}\right\} $$ is an isomorphism when the addition and multiplication of $\ X/\sim\ $ are defined in the obvious way.

Addendum: It is also the case that $\ \varphi\ $ is the only isomorphism between $\ \mathbb{Z}\ $ and any subset of $\ X/\sim\ $. It has occurred to me that if this question were asked as part of a test or an exercise in a course, the person asking the question might be expecting you to prove this uniqueness as well.