I came across this proof of the countability of $\mathbb{Q}$ in a textbook:      Each rational can be expressed as a fraction $a/b$, where $a$ and $b$ are integers. We know that the set $\{(a,b) : a,b\in\mathbb{Z}\}$ is countable, thus $\mathbb{Q}$ is countable.    My question is this: Clearly each rational can be uniquely expressed in lowest terms as a fraction $a/b$. However, each fraction can be expressed in a multitude of different ways that are in the set $S=\{(a,b) : a,b\in\mathbb{Z}\}$, specifically $(na)/(nb)$, where $n\in\mathbb{Z}$. Thus we only know from this that $|\mathbb{Q}|\leq|S|$, and not that $|\mathbb{Q}|=|S|$. Because of this, does this mean that we are assuming CH?   *As a side note: I'm not questioning whether or not the rationals are countable, I'm completely sure that they are. I am just asking if this proof is assuming CH or not.   Thank you all in advance for the help.
I remember my professor in college challenging me with this question, which I failed to answer satisfactorily: I know there exists a bijection between the rational numbers and the natural numbers, but can anyone produce an explicit formula for such a bijection?
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