I have a trouble understanding the representations of objects...

Saying we have $$\cal{R}:\mathbb{Q}^+ \longrightarrow~~\mathbb{N}\times\mathbb{N}$$defined by $\cal{R}\left(\dfrac{a}{b}\right) = (a, b)$, where $a,b\in\mathbb{N}$ and $b\neq0$

Is $\cal{R} $ a function or not?

Question

I have a trouble understanding the representations of objects...



Saying we have $$\cal{R}:\mathbb{Q}^+ \longrightarrow~~\mathbb{N}\times\mathbb{N}$$defined by $\cal{R}\left(\dfrac{a}{b}\right) = (a, b)$, where $a,b\in\mathbb{N}$ and $b\neq0$


Is $\cal{R} $ a function or not?

geerky42 · Answer

What so hard for me to understand is that we know that $\dfrac{1}{2} = \dfrac{2}{4}$, so we could say that $\cal{R}$ is not function because $\cal{R} \left( \dfrac{1}{2}\right) = (1,2)$ and $(2,4)$.


But $\dfrac{2}{4}$ doesn't necessary exist in $\mathbb{Q}^+$, since it already has $\dfrac{1}{2}$, because, well, $\dfrac{2}{4}$ is simplified to $\dfrac{1}{2}$ right? 


What I said is probably sounds dumb, but it is ambiguous to me...

zarkon · Answer

it is not well defined because of the reason you stated 1/2 and 2/4 give different outputs

Also, 2/4 is in  $\mathbb{Q}^+$