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Math Help - Equivalence Class Representatives

  1. #1
    is up to his old tricks again! Jhevon's Avatar
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    Equivalence Class Representatives

    Okay, so this question seemed easy to me earlier, but now i'm getting paranoid about it.

    Problem:

    Define a relation \sim on the set \mathbb{N} of natural numbers by

    a \sim b iff a = b \cdot 10^k for some k \in \mathbb{Z}

    (b) Give a complete set of equivalence class representatives


    Things That Might Come In Handy:

    I suppose whoever helps me with this will know what an equivalence class is, so i will tell you about equivalence class representatives.

    let S be a set which we have defined an equivalence relation on. The set of equivalence class representatives is the subset of S containing precisely one element from each equivalence class.

    so, for example. lets say i have the set \mathbb{Z}, and my relation on this set is

    for a,b \in \mathbb{Z}, a \sim b iff a and b have the same parity.

    clearly this partitions the integers into the set of even integers and the set of odd integers. so if e is any even integer, its equivalence class is [e] = \{ 0,~ \pm 2,~ \pm 4, \cdots \} and if o is any odd integer, its equivalence class is given by [o] = \{ \pm 1,~ \pm 3,~ \pm 5, \cdots \}

    for the set of equivalence class representatives, we would simply choose any one even number and any one odd number, so \{ 0,1 \} suffices.


    What I have tried:

    I realize for any a \in \mathbb{N}, [a] = \{a \cdot 10^k | k \in \mathbb{N} \}

    but it gets more complicated. if a happens to be a multiple of 10, then for some of the k's in the above set, -k also works, but not all. secondly, what would i choose for the set of representatives. i was thinking about choosing \mathbb{N}, but this will cause repeats, as for example, 1, 10, 100, 1000 etc will be in there when all are in [1] and [10] and [100] etc. how would i define my set here?
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  2. #2
    is up to his old tricks again! Jhevon's Avatar
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    i think \{ n ~|~ n \in \mathbb{N} \mbox{ and } n \mbox{ is not a multiple of } 10 \} works.

    does it?
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