1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Prove that Q under addition is not isomorphic to R+

  1. Oct 20, 2004 #1
    How do I prove that Q under addition is not isomorphic to R+ under multiplication?
     
  2. jcsd
  3. Oct 20, 2004 #2

    matt grime

    User Avatar
    Science Advisor
    Homework Helper

    They cannot be isomorphic as groups because they are not even in bijective correspondence as <insert one word to get the answer>
     
  4. Oct 20, 2004 #3
    Isn't it f(x) = exp(x) a bijection between Q and R+?
     
  5. Oct 20, 2004 #4

    jcsd

    User Avatar
    Science Advisor
    Gold Member

    No an isomorphism must be onto.
     
  6. Oct 20, 2004 #5
    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?
     
  7. Oct 20, 2004 #6

    jcsd

    User Avatar
    Science Advisor
    Gold Member

    If r is irrational is r in Q? is er in R+?
     
  8. Oct 20, 2004 #7
    You're totally right. Then is there any way to show that Q and R+ are not isomorphic?
     
  9. Oct 20, 2004 #8
    I think I know the answer. If I say that any map between Q and R+ is not onto, is that enough?
     
  10. Oct 20, 2004 #9

    jcsd

    User Avatar
    Science Advisor
    Gold Member

    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).
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Prove that Q under addition is not isomorphic to R+
  1. Proving an Isomorphism (Replies: 6)

  2. Proving isomorphism (Replies: 12)

  3. Prove not isomorphic? (Replies: 2)

Loading...