Is there a way of writing $\mathbb{Q}$ in terms of $\mathbb{Z}^2$ in a simple expression?
My idea is to construct $\mathbb{Q}$ a quotient set of $\mathbb{Z}^2$, where $\mathbb{Z}$ is the set of integers. To form $\mathbb{Q}$, we could start with the Cartesian product $\mathbb{Z}\times (\mathbb{Z}\backslash\{0\})$ which is the set of all ordered pairs $(p, q)$ such that $p$ is an integer and $q$ is a non-zero integer. These pairs would correspond to the fractions $p/q$.
However, this set contains redundancies because the same rational number can be represented in different ways (for example, $1/2$ is the same as $2/4$). To obtain a set that corresponds exactly to the rational numbers, we could identify pairs $(p, q)$ and $(p', q')$ if they represent the same rational number. This is the same as saying that $(p, q)$ and $(p', q')$ are identified if $pq' = p'q$.
The result would then be a quotient set $(\mathbb{Z}\times (\mathbb{Z}\backslash\{0\}))/\text{~}$, where ~ is the equivalence relation defined by $(p, q) \text{~} (p', q')$ if and only if $pq' = p'q$. This quotient set is isomorphic to $\mathbb{Q}$, the set of rational numbers.
Is this the best way to do it? It is not written explicitly in terms of $\mathbb{Z}^2$, I had to somehow deconstruct it, but I wonder if there is a better way to write it. From here, for example, we see that $\mathbb{Q} \cong \mathbb{Z^2} $ can be ambiguous.