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Equivalence Relations

  1. Aug 15, 2011 #1
    1. The problem statement, all variables and given/known data

    Deciede if the following are equivalence relations on Z. If so desribe the eqivalence classes
    i) a[itex]\equiv[/itex] b if [itex]\left|a\right|[/itex] = [itex]\left|b\right|[/itex]
    ii) a[itex]\equiv[/itex] b if b=a-2
    2. Relevant equations



    3. The attempt at a solution

    i) [itex]\left|a\right|[/itex] = [itex]\left|a\right|[/itex] so its reflexive

    [itex]\left|a\right|[/itex] = [itex]\left|b\right|[/itex] is equivalent to [itex]\left|b\right|[/itex] = [itex]\left|a\right|[/itex] so its symmetric

    [itex]\left|a\right|[/itex] = [itex]\left|b\right|[/itex] and [itex]\left|b\right|[/itex] = [itex]\left|c\right|[/itex] then [itex]\left|a\right|[/itex] = [itex]\left|c\right|[/itex] for all values a,b and c elemets of Z so its transitive.

    Are there infinite equivalence classes??


    ii) a=a so its reflexive
    b=a-2 [itex]\neq[/itex] a=b-2 so its not symetric, am i right in thinking this?
    Thanks for reading
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Aug 15, 2011 #2

    vela

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    Yes. Can you describe them? Simply listing a few to show the pattern would be sufficient.
    a=a-2?
    Yes.
     
  4. Aug 15, 2011 #3

    SammyS

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    For (i):
    What elements(s) of Z is/are equivalent to 3?
    What elements(s) of Z is/are equivalent to 7?
    What elements(s) of Z is/are equivalent to 0?
    What elements(s) of Z is/are equivalent to -5?
    ...​

    For (ii):
    This relation is not transitive either.​
     
  5. Aug 15, 2011 #4
    Thanks for the replies.
    So i need to say there are infinity equivalent classes such as -3 equivalent to 3; -5 equivalent to 5 or 10 is equivalent to -10 under the relation.

    for ii) i only need to show 1 of the 3 properties doesnt hold, right? or should i show whether all 3 hold or not just for clarity?
     
  6. Aug 15, 2011 #5

    vela

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    Basically, yes, though your instructor may cringe at your grammar. ;)

    The equivalence classes are subsets consisting of all elements that are equivalent to each other. So in this case, they'd be {0}, {1,-1}, {2,-2}, and so on.
    Right. You need to show only one requirement doesn't hold to rule out the relation being an equivalence relation.
     
  7. Aug 15, 2011 #6
    Grammer isnt a strong point of mine :)
    Thanks a mill
     
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