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Math Help - natural numbers...

  1. #1
    zee
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    Unhappy natural numbers...

    THere is a proof that integers can be constructed from the natural numbers
    by identifying Z witht he quotient set NxN/~.
    Can anyone help with this proof??
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  2. #2
    Super Member Rebesques's Avatar
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    Re: natural numbers...

    You can define a relation R on \mathbb{N}\times\mathbb{N} by (a,b)R(c,d)\Leftrightarrow a+d=c+b.
    Prove that R is an equivalence relation (how?).
    The equivalence classes for R now form a group, which is isomorphic to the integers, with the (class independent- prove it for fun) operation [(a,b)]+[(c,d)]=[(a+b,c+d)].
    Last edited by Rebesques; December 27th 2014 at 06:47 AM.
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  3. #3
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    Re: natural numbers...

    Note that we can think of the "natural numbers" as a subset of the integers, defined in this way, by associating the natural number n with the class containing the pair (a, a+ n). The number "0" is then the class containing (n, n) for all natural numbers n.
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