Prove that Q under addition is not isomorphic to R+

[afirican]

How do I prove that Q under addition is not isomorphic to R+ under multiplication?

 

[matt grime]

They cannot be isomorphic as groups because they are not even in bijective correspondence as <insert one word to get the answer>

 

[afirican]

Isn't it f(x) = exp(x) a bijection between Q and R+?

 

[jcsd]

Isn't it f(x) = exp(x) a bijection between Q and R+?

No an isomorphism must be onto.

 

[afirican]

Why f:Q -> R+, f(x) = exp(x) is not onto?
For all r of R+, there exists r' = lnr in Q such that r = exp(lnr) = exp(r') = f(r'). Where do I go wrong?

 

[jcsd]

If r is irrational is r in Q? is e^r in R⁺?

 

[afirican]

You're totally right. Then is there any way to show that Q and R+ are not isomorphic?

 

[afirican]

I think I know the answer. If I say that any map between Q and R+ is not onto, is that enough?

 

[jcsd]

Yes, you just need to look at the two sets Q and R⁺ to see that the two groups cannot be isomorphic (as Matt grime indicated).