Results 1 to 2 of 2

Thread: Binary Structures and Mapping

  1. #1
    Dec 2010

    Binary Structures and Mapping

    I am having great difficulty conceptualizing binary structures and mapping from one to the other.

    Here's my problem:

    The map \phi : \mathbb{Z} \rightarrow \mathbb{Z} defined by \phi(n) = n + 1 for n \in \mathbb{Z} is one-to-one and onto \mathbb{Z}. Give the definition of a binary operation * on \mathbb{Z} such that \phi is an isomorphism mapping.

    <\mathbb{Z},\cdot> with <\mathbb{Z},*>.

    So I have <\mathbb{Z},\cdot>. This means that for members a and b, this structure is a \cdot b and d \cdot a, right? And I'm trying to map this to an unknown structure? I don't understand what exactly I'm doing.

    What is that \phi(n) = n + 1 function? I assume it takes a number in \mathbb{Z} and maps it to a number in \mathbb{Z}, specifically one more than the input number. But what does that have to do with my structures? What am I trying to find?

    In utter confusion, I took <\mathbb{Z},\cdot> and figured that for the inputs of a=2 and b=3, the answer would be 6 since 2*3=6. So I take that 6... add one to it from \phi, and that gives me 7... is that 7 the value that is supposed to correspond with the second structure, <\mathbb{Z},*>? So is my task to find a function * that takes two variables a and b to make 7, or more abstractly, ab+1??

    Any help = appreciated.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Mar 2011

    Re: Binary Structures and Mapping

    what you are being asked to do is FIND a definition for * so that φ(ab) = φ(a)*φ(b).

    the fact that φ is 1-1 and onto already is the "iso" part....ensuring that φ(ab) = φ(a)*φ(b) is the "morphism" part.

    suppose that a*b is defined as ab+1...does this definition of * fit the requirements?

    (your question is a little vague...since no other information is given, i am assuming you aren't requiring that * be associative, or have an identity).
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Isomorphic binary structures
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: Sep 21st 2011, 10:46 PM
  2. Replies: 6
    Last Post: Jul 15th 2011, 05:11 PM
  3. Complex open mapping & conformal mapping problems.
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: Feb 22nd 2011, 08:26 AM
  4. Structures
    Posted in the Discrete Math Forum
    Replies: 0
    Last Post: Oct 27th 2008, 11:45 PM
  5. Help with Structures
    Posted in the Algebra Forum
    Replies: 5
    Last Post: Jun 7th 2006, 10:53 AM

Search Tags

/mathhelpforum @mathhelpforum