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How to prove two binary structures are not isomorphic

  1. Apr 19, 2012 #1
    1. The problem statement, all variables and given/known data

    Let * be the binary operation defined on Z by

    m*n= the smaller of m and n, if m =/= n, or

    m or n, if m=n.

    Show that the binary structures <Z, +> and <Z,*> are not isomorphic.


    3. The attempt at a solution

    Proof by contradiction:

    Let F be a map from <Z,+> to <Z,*>. Then F is onto, one-to-one, and F(x+y)=F(x)*F(y).

    1. F is onto.
    If F(x)=F(y), then x=y, so F is onto.

    2. F is one-to-one.

    For every F(z) in <Z,*>, there's a F inverse of F(z)=z in <Z,+>.

    3. F(x+y)=F(x)*F(y).

    Suppose x<y<z<w, then

    F(x+y) =/= F(x)*F(y) = F(x)
    F(x+z) =/= F(x)*F(y)= F(x)
    F(x+w) =/= F(x)* F(w)= F(x).

    Therefore <Z,+> and <Z,*> are not isomorphic.


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    Any help is appreciated!
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
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