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Finding the no of equivalent classes

  1. Oct 21, 2010 #1
    1. The problem statement, all variables and given/known dataplease consider the relation over the set of integers Z , given by m=n mod (p) where p is a positive integer . Prove that it is an equivalent relation.
    find the elements in the equivalent class of m.
    find the no of such equivalent classes.

    2. Relevant equations



    3.equivalent relation-proved by showing reflexivity, symmetry and transitivity.
    the equivalent class of m consists of elements of the type
    m +k p where k = 0,+/-1,+/-2......
    but i am not able to think how to find the no of such classes. any help will be highly appreciated.
     
  2. jcsd
  3. Oct 21, 2010 #2

    tiny-tim

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    hi commutator! :smile:

    (have a ± :wink:)
    that's right! :smile:

    ok, when you're stuck it's often useful to try an easy example …

    if p = 10, what are the equivalence classes? :wink:
     
  4. Oct 21, 2010 #3
    thanks . that was really encouraging.

    for p=10, i got something like [-1]=(.....,-11,-21-1,9,19,29......)
    [0]=(....-10,-20.....0,10,20,.....)
    [1]=(...-29,-19....1,11,21,31...]
    interestingly , [11] is coming same as [1]. [-1] is coming the same as [9].so i think there are 10 distinct classes, which are repeating over and over. then the answer has got to be 10. ie. p. but how do i sketch a formal proof ? by induction?
     
  5. Oct 21, 2010 #4

    tiny-tim

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    hi commutator! :smile:
    that's right, there's exactly p classes

    the formal proof isn't the problem …

    the problem is the formal definition

    once you have that, the proof is obvious

    eg your definition {...-29,-19....1,11,21,31...} is only a list, so that's not helpful, but your earlier definition (I'm expanding it a little) [m] = {m +kp : k in Z} should do it :smile:

    (btw, always use squirly brackets for listing the elements of sets :wink:)
     
  6. Oct 21, 2010 #5
    thanks a lot !!!!
     
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