How to prove two binary structures are not isomorphic

  • #1

Homework Statement



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.


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!

Homework Statement





Homework Equations





The Attempt at a Solution

 

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