Results 1 to 7 of 7

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

  1. #1
    Newbie
    Joined
    Jul 2011
    Posts
    5

    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?

    -----------------------
    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
    Last edited by mr fantastic; July 14th 2011 at 03:51 AM. Reason: Re-titled.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Mar 2010
    From
    Florida
    Posts
    3,093
    Thanks
    5

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

    Quote Originally Posted by kashnex View Post
    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.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Jul 2011
    Posts
    5

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

    Quote Originally Posted by dwsmith View Post
    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
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    21

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

    Quote Originally Posted by kashnex View Post
    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).
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Newbie
    Joined
    Jul 2011
    Posts
    5

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

    Quote Originally Posted by Drexel28 View Post
    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
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Senior Member
    Joined
    Feb 2008
    Posts
    410

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

    Quote Originally Posted by kashnex View Post
    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.
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Newbie
    Joined
    Jul 2011
    Posts
    5

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

    Quote Originally Posted by hatsoff View Post
    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.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 0
    Last Post: December 17th 2011, 06:05 PM
  2. Binary Structures and Mapping
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: September 21st 2011, 10:43 PM
  3. Isomorphic binary structures
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: September 21st 2011, 09:46 PM
  4. Binary Operation
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: August 4th 2010, 10:39 AM
  5. binary operation
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: September 4th 2008, 02:48 PM

Search Tags


/mathhelpforum @mathhelpforum