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

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

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

    What is that $\displaystyle \phi(n) = n + 1$ function? I assume it takes a number in $\displaystyle \mathbb{Z}$ and maps it to a number in $\displaystyle \mathbb{Z}$, 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 $\displaystyle <\mathbb{Z},\cdot>$ 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 $\displaystyle \phi$, and that gives me 7... is that 7 the value that is supposed to correspond with the second structure, $\displaystyle <\mathbb{Z},*>$? So is my task to find a function $\displaystyle *$ that takes two variables $\displaystyle a$ and $\displaystyle b$ to make 7, or more abstractly, ab+1??

    Any help = appreciated.
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  2. #2
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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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