The set of rational numbers is defined as $\mathbb{Q} = \left\lbrace \frac{a}{b} \mid a, b \in \mathbb{Z} \land b \neq 0 \right\rbrace$.
This apparently means that $\frac{1}{2}$ and $\frac{2}{4}$ are distinct two elements of the set $\mathbb{Q}$. And similarly, every $\frac{0}{n}$ for all $n \in \mathbb{Z} \setminus \left\lbrace 0 \right\rbrace$ are also distinct elements of $\mathbb{Q}$. Is this right?
And if that is right, for a function $f : \mathbb{Z} \to \mathbb{Q}$ to be a bijection, there has to exists distinct $n, m \in \mathbb{Z}$ which satisfy $f(n) = \frac{0}{1}$ and $f(m) = \frac{0}{2}$, and of course $n \neq m$. Is this right as well?
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$\begingroup$By virtue of the fact that the definition is not in terms of ordered pairs but rather in terms of fractions, all of the rules of fractions apply. In particular, for any $a\in\mathbb{Z}$, $b\in\mathbb{Z}-\{0\}$, and $c\in\mathbb{Z}-\{0\}$ we have $$\frac{ac}{bc}=\frac{a}{b}$$ and, as mentioned in the comments, $$\frac{a}{b}=\frac{c}{d}$$ if and only if $ad-bc=0$.
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