What is the proper notation when defining certain functions?
For example, consider the function $\phi:\mathbb{Q} \rightarrow \mathbb{Z}\times\mathbb{N}$ defined by $\phi(\frac{p}{q})=(p,q)$ for any $\frac{p}{q}\in\mathbb{Q}$ such that p and q have no common divisor except 1. The way this function is defined relies on how the set $\mathbb{Q}$ is defined, that is, $\mathbb{Q}=\{\frac{p}{q}:p\in\mathbb{Z},q\in\mathbb{N}\}$.
I have seen that sets can be defined in general as $\{\sigma(n_1,n_2,...,n_k):n_1\in N_1,n_2\in N_2,...,n_k\in N_k\}$, where $\sigma$ is some sort of formula that depends of the sets $N_1,N_2,...,N_k$. Can we somehow define this function, not by the form of the elements of $\mathbb{Q}$ but rather an arbitrary element, say r, such that we can write the images of $\phi$ as $\phi(r)$? It always confuses me that when these sort of functions are defined this way, the function depends of several variables instead of one like the one previously mentioned.