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Math Help - Equivalence relation and Functions

  1. #1
    Junior Member
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    Equivalence relation and Functions

    Define Z = \{((x_{1},y_{1}),(x_{2},y_{2}))  \epsilon  (N \times N) \times (N \times N): x_{1}+y_{2}= y_{1} +x_{2}\}

    show that Z is an equivalence relation on N x N

    Show that

    N = \{([(x,y)]_{Z},[(y,x)]_{Z}) :x,y  \epsilon  N \}

    is a function

    Show that
    A= \{(([x_{1},y_{1})]_{Z},[(x_{2},y_{2})]_{Z}), [(x_{1}+x_{2},y_{1}+y_{2})]_{Z}): x_{1},x_{2},y_{1},y_{2} \epsilon N \}
    is a function

    I'm just having trouble setting these problems up
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  2. #2
    MHF Contributor
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    Why don't you start by writing the three properties of an equivalence relation? For example, reflexivity says \forall p\in\mathbb{N}\times\mathbb{N}\,(p,p)\in Z, or \forall x,y\in\mathbb{N}\,((x,y),(x,y))\in Z. For this particular definition of Z, this means \forall x,y\in\mathbb{N}\,x+y=x+y, which is obviously true.
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