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Math Help - Binary Structures and Mapping

  1. #1
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    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.
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  2. #2
    MHF Contributor

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    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).
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