A function $ f:\mathbb{Q}_{>0}\longrightarrow\mathbb{Q} $ has the following property: $$ f(xy)=f(x)+f(y),\quad x,y\in\mathbb{Q}_{>0} $$ a) Demonstrate that there are no injective functions with this property. b) Do exist surjective functions having this property?
Problem
Source: Romanian National Olympiad 2017, grade x, p.2
Tags: function, algebra
26.08.2019 08:22
26.08.2019 08:27
Revise this NikoIsLife wrote: For any rational number $x$, suppose that $x=p_1^{e_1}p_2^{e_2}p_3^{e_3}\cdots$ where $e_1,e_2,e_3,\ldots$ are integers.
26.08.2019 08:33
CatalinBordea wrote: Revise this NikoIsLife wrote: For any rational number $x$, suppose that $x=p_1^{e_1}p_2^{e_2}p_3^{e_3}\cdots$ where $e_1,e_2,e_3,\ldots$ are integers. I'm sorry if I wasn't clear enough. I changed the wording so it would become less confusing.
26.08.2019 09:16
Are you sure that the codomain your function at b) is included in $ \mathbb{Q} ? $
26.08.2019 09:24
CatalinBordea wrote: Are you sure that the codomain your function at b) is included in $ \mathbb{Q} ? $ Yes it is, because $\mu_{p_k}(x)$ is always integer for all $k$ and $x$, which means that $f(x)$ is always a rational number.
26.08.2019 09:35
I thought it is an infinite sum, which is not. Now I understand it.