How to prove two binary structures are not isomorphic

[nontradstuden]

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!

 

[mighty2000]

Hi there,

Your proof seems to be on the right track, but there are a few issues that need to be addressed. Here are some suggestions for improving your proof:

1. When proving that F is onto, you need to show that for every element in <Z,*>, there exists an element in <Z,+> that maps to it under F. Right now, your proof only shows that F is one-to-one.

2. In your proof that F is one-to-one, you should specify that F is only one-to-one when restricted to the set of integers. Otherwise, it's possible that F is one-to-one on a larger set of numbers, but not on the integers.

3. Your proof that F(x+y) =/= F(x)*F(y) is unclear. It's not clear how you're using the fact that x<y<z<w to show that these two expressions are not equal. Also, it's not clear why you're only considering the case where x<y<z<w. You need to show that F(x+y) is not equal to F(x)*F(y) for all possible values of x and y in <Z,+>.

Hope this helps! Keep up the good work.