Suppose we have a set $S$ of real numbers,such that 


$ n \neq \Pi\_{m \in S-{n}} m^{r\_m}$....... $\forall n\in S$ where $r\_{m}$ is a rational number....  

 The set of prime number is an example of such a ser...now my question is that...Is such a set necessarily countable?



a question about subset of R
Since we are talking about rational exponents, I assume it makes sense to restrict to positive real numbers. If I got it right the question is whether a set $S$ where no element is a product of some others to rational powers is necessarily countable. 


One can show non-constructively that the answer is no, that is, that there exist uncountable sets with this property. This is because the condition on $S$ is equivalent to $\{\log x: x\in S\}$ being linearly independent over the rational numbers. 


Since there is a basis for $\mathbb{R}$ as a vector space over $\mathbb{Q}$, and such a basis is necessarily uncountable, there is an uncountable set $S$ with the given property. 


