Is the relation a function

Question

I'm trying to determine if the relation $\{(\frac{a}{b}, a-b) | a,b \in \Bbb {Z}, b \neq 0\}$ is a function from $\Bbb {Q}$ to $\Bbb {Z}$.

I know that a relation is a function from A to B if dom(f)=A and ran(f) $\subseteq$ B. And if for every $a \in A$ there is exactly one $b \in B$ such that $(a,b) \in f$.

But I'm still confused on how to go about proving or disproving this.

Answer 1

Not true because $\dfrac{1}{2} = \dfrac{2}{4}$ but $1-2 = -1$ while $2-4 = -2$

Answer 2

It is not. Your relation will include $(\frac 1 2,-1)$ and $(\frac 2 4, -2) = (\frac 1 2, -2)$ implying that $\frac 1 2$ is mapped to both $-1$ and $-2$, which violates the definition of a function.