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Math Help - Equivalence Relations: Symbol Confusion

  1. #1
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    Equivalence Relations: Symbol Confusion

    "THE" QUESTION...
    Let A = Z (all integers)
    and let R be the relation defined on A by a R b if and only if a|b.
    Is R an equivalance relation?

    "MY" QUESTION...
    I'm thrown off by the | symbol - is this saying that a is dividable by b? Which would include the following...
    {(1,1),(2,1),(2,2),(3,1),(3,3),(4,1),(4,2),(4,4).. .}

    Any help would be greatly appreciated!
    Doug
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  2. #2
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    The relation R on your question is not an equivalence relation, rather, it is a "partial order" relation.

    If R be an equivalence relation, it should satisfy following three conditions.

    1. Reflexive ( aRa )
    2. Symmetry (if aRb, then bRa)
    3. Transitive (if aRb and bRc, then aRc)

    (aRb means aRb = TRUE)

    Your question satisfies condition 1 and 3 but it does not satisfy the condition 2 ( For instance, 3|6 is true, but 6|3 not ).
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  3. #3
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    Thank you. I take it by your answer that you agree a|b means a is divisible by b (or at least b divisible by a in your example of 3,6)?

    I haven't seen this symbol (pipe on the keyboard) elsewhere in our textbook, and wasn't sure of its meaning.

    Thanks again for your quick and thorough answer!
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  4. #4
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    Quote Originally Posted by dstclair View Post
    "THE" QUESTION...
    Let A = Z (all integers)
    and let R be the relation defined on A by a R b if and only if a|b.
    Is R an equivalance relation?
    This is a really TRICK question!
    Does zero divide zero?
    If not, can it be reflexive?
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  5. #5
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    Quote Originally Posted by Plato View Post
    This is a really TRICK question!
    Does zero divide zero?
    If not, can it be reflexive?
    You are right. My mistake
    "|" is not reflexive over all integers because 0|0 is not defined.
    Last edited by vagabond; November 2nd 2008 at 07:34 PM.
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