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

**kashnex** · July 13th 2011, 07:41 PM

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

Bundle of thanks in advance.
Regards

**dwsmith** · July 13th 2011, 07:46 PM

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

Bundle of thanks in advance.
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.

**kashnex** · July 13th 2011, 07:49 PM

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

**Drexel28** · July 13th 2011, 08:04 PM

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).

**kashnex** · July 13th 2011, 08:31 PM

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.

please please please

**hatsoff** · July 14th 2011, 01:49 AM

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

Bundle of thanks in advance.
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.

**kashnex** · July 15th 2011, 04:11 PM

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.

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