Is $f:\mathbb{Q^*} \rightarrow \mathbb{Q}$ by $f(\frac{a}{b}) = \frac{\max{(a,b)}}{\min{(a,b)}}$ a function?

Question

Suppose that the relation $f:\mathbb{Q^*} \rightarrow \mathbb{Q}$ by $$ f \Bigl(\frac{a}{b} \Bigr) = \frac{\max{(a,b)}}{\min{(a,b)}} $$ is defined. Then is $f$ a function? If so, how would we prove that $f$ is a function?

Answer 1

You'd need to prove that if $\frac{a}{b}=\frac{c}{d}$ then $$\frac{\max(a,b)}{\min(a,b)}=\frac{\max(c,d)}{\min(c,d)}$$

Then it depends on what $a,b$ are allowed. We know that $a,b\neq 0$, but can they have a common divisor? Can $b$ be negative? Or are we taking some "usual" representation for each rational - for example, $a,b$ relatively prime with $b>0$.