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Math Help - Equivalence relations and classes.

  1. #1
    Super Member Showcase_22's Avatar
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    Equivalence relations and classes.

    let n be a fixed non-negative integer, and define a relation ~ on the set of integers by setting a~b iff a-b is an integer multiple of n. Show that ~ is an equivalence relation and describe the corresponding equivalence classes when:

    a). n=2
    b). n=1
    c). n=0
    a~b if a-b=kn

    Reflexivity: a-a=0=0n and b-b=0=0n so reflexivity holds.

    Symmetry: a~b \Rightarrow a-b=kn
    Therefore b-a=-kn so symmetry holds.

    Transitivity a~b \Rightarrow a-b=kn
    b~c \Rightarrow b-c=qn

    a-b+b-c=(k+q)n so a-c=(k+q)n so transitivity holds.

    a). a-b=2q. This splits \mathbb{Z} into all the odd and even numbers since even-even=even and odd-odd=odd.

    b). a-b=q
    \mathbb{Z} is not split. There is one one class- \mathbb{Z} itself.

    c). a-b=0
    this implies that a=b. This seems to suggest that equivalence classes cannot exist.

    Are these right? c). especially seems a little curious.
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  2. #2
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    Quote Originally Posted by Showcase_22 View Post

    c). a-b=0
    this implies that a=b. This seems to suggest that equivalence classes cannot exist.

    Are these right? c). especially seems a little curious.
    It implies that all numbers are in an equivalence class of their own.

    Bobak
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