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Thread: Well-defined map

  1. #1
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    Well-defined map

    How can I show that the map, here Z= integers

    \phi: Z/3Z /times Z/3Z ---> Z/3Z defined by phi(x,y) = xy is well-defined.
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    Quote Originally Posted by peteryellow View Post
    How can I show that the map, here Z= integers

    \phi: Z/3Z /times Z/3Z ---> Z/3Z defined by phi(x,y) = xy is well-defined.
    If $\displaystyle [x_1]=[x_2]$ and $\displaystyle [y_1]=[y_2]$ then it means $\displaystyle x_1\equiv x_2(\bmod 3)$ and $\displaystyle y_1\equiv y_2(\bmod 3)$.
    Therefore, $\displaystyle x_1y_1\equiv x_2y_2(\bmod 3)$ and so $\displaystyle [x_1y_1] = [x_2y_2]$.
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