# Thread: Existence of binary operation such that two algebraic structures are isomorphic.

1. ## Existence of binary operation such that two algebraic structures are isomorphic.

Here is a very tricky question that I am trying to solve since last three days but until now I couldn't find the solution. Can anyone be of some help to solve this question?

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Does a binary operation $\diamond$ exist such that $\left\langle N, +\right\rangle$ and $\left\langle Z, \diamond\right\rangle$ are isomorphic? Prove your answer.
NOTE: $N$ is the set of natural numbers, $Z$ is the set of integers and + is the addition operation. Two algebraic structures $\left\langle A, \times\right\rangle$ and $\left\langle B, \otimes\right\rangle$ are isomorphic if and only if a bijection $h:A \rightarrow B$ exists such that $h(x \times y)= h(x)\otimes h(y)$
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I would appreciate if full proof of this problem can be written

Regards

2. ## Re: A Tricky Question ... Any help to solve it ...!

Originally Posted by kashnex
Here is a very tricky question that I am trying to solve since last three days but until now I couldn't find the solution. Can anyone be of some help to solve this question?

-----------------------
Does a binary operation $\diamond$ exist such that $\left\langle N, +\right\rangle$ and $\left\langle N, \diamond\right\rangle$ are isomorphic? Prove your answer.
NOTE: N is the set of natural numbers, Z is is the set of integers and + is the addition operation. Two algebraic structures $\left\langle A, \times\right\rangle$ and $\left\langle B, \otimes\right\rangle$ are isomorphic if and only if a bijection $h:A \rightarrow B$ exists such that $h(x \times y)= h(x)\otimes h(y)$
------------------------

I would appreciate if full proof of this problem can be written

Regards
In general, you are supposed to make some kind of attempt no matter how wrong it may be instead of just ask for others to do your work.

3. ## Re: A Tricky Question ... Any help to solve it ...!

Originally Posted by dwsmith
In general, you are supposed to make some kind of attempt no matter how wrong it may be instead of just ask for others to do your work.
The problem is that I have no time because I am going to take entry exam on Monday, but didn't understand this question that how I can solve it. Please if you can be some help please suggest me or give me some pointers that enables me to solve it.

regards

4. ## Re: A Tricky Question ... Any help to solve it ...!

Originally Posted by kashnex
The problem is that I have no time because I am going to take entry exam on Monday, but didn't understand this question that how I can solve it. Please if you can be some help please suggest me or give me some pointers that enables me to solve it.

regards
Think about what makes the natural numbers complete (induction).

5. ## Re: A Tricky Question ... Any help to solve it ...!

Originally Posted by Drexel28
Think about what makes the natural numbers complete (induction).
So poor in understanding it that how i can apply the completeness (induction).
Kindly help me out by giving complete solution. I would appreciate that.

6. ## Re: A Tricky Question ... Any help to solve it ...!

Originally Posted by kashnex
Here is a very tricky question that I am trying to solve since last three days but until now I couldn't find the solution. Can anyone be of some help to solve this question?

-----------------------
Does a binary operation $\diamond$ exist such that $\left\langle N, +\right\rangle$ and $\left\langle Z, \diamond\right\rangle$ are isomorphic? Prove your answer.
NOTE: $N$ is the set of natural numbers, $Z$ is the set of integers and + is the addition operation. Two algebraic structures $\left\langle A, \times\right\rangle$ and $\left\langle B, \otimes\right\rangle$ are isomorphic if and only if a bijection $h:A \rightarrow B$ exists such that $h(x \times y)= h(x)\otimes h(y)$
------------------------

I would appreciate if full proof of this problem can be written

Regards
This is quite inappropriate. You should never demand a full proof. However, since I feel your pain about the exam, I will say this much:

Since $\mathbb{N},\mathbb{Z}$ are equinumerous, we may let $\varphi:\mathbb{N}\to\mathbb{Z}$ be any bijection, and define the binary operation $\diamond:\mathbb{Z}\times\mathbb{Z}\to\mathbb{Z}$ by $\varphi(m)\diamond\varphi(n)=\varphi(m+n)$ for all $m,n\in\mathbb{N}$. Then it follows immediately that $\varphi$ is an isomorphism. It remains to prove that $\diamond$ is well-defined.

7. ## Re: A Tricky Question ... Any help to solve it ...!

Originally Posted by hatsoff
This is quite inappropriate. You should never demand a full proof. However, since I feel your pain about the exam, I will say this much:

Since $\mathbb{N},\mathbb{Z}$ are equinumerous, we may let $\varphi:\mathbb{N}\to\mathbb{Z}$ be any bijection, and define the binary operation $\diamond:\mathbb{Z}\times\mathbb{Z}\to\mathbb{Z}$ by $\varphi(m)\diamond\varphi(n)=\varphi(m+n)$ for all $m,n\in\mathbb{N}$. Then it follows immediately that $\varphi$ is an isomorphism. It remains to prove that $\diamond$ is well-defined.

Thank you very much for this reply.
My apologies for asking a full proof of that..... however; i was pretty desperate of understand the problem. Thank you again for providing me pointer. I am trying hard to solve this problem.