How do I prove that Q under addition is not isomorphic to R+ under multiplication?
They cannot be isomorphic as groups because they are not even in bijective correspondence as <insert one word to get the answer>
Isn't it f(x) = exp(x) a bijection between Q and R+?
No an isomorphism must be onto.
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?
If r is irrational is r in Q? is er in R+?
You're totally right. Then is there any way to show that Q and R+ are not isomorphic?
I think I know the answer. If I say that any map between Q and R+ is not onto, is that enough?
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).
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