Finding Equivalence Class

  1. 1. The problem statement, all variables and given/known data
    Find the equivalence class [2] for the following equivalence relations:

    a) R: Z <-> Z, where xRy, iff |x| = |y|

    b) T: N <-> N, where xTy, iff xmod4 = ymod4

    N means natural numbers etc...there wasnt the correct symbols in the latex reference

    2. Relevant equations

    ??

    3. The attempt at a solution

    Ok so I know how to do the b) part, because we had examples at the class, its:

    [0] = {0,4,8,12,...}
    [1] = {1,5,9,13,...}
    [2] = {2,6,10,14,...}

    so the answer is [2] = {2,6,10,14,...} right?

    but i dont know how i start to build it when i have |x| = |y|
    its probably something very easy and i just dont get it for some reason
     
  2. jcsd
  3. George Jones

    George Jones 6,357
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    Suppose x is given but unknown, and that |x| = |y|. What can y equal in terms of the given x?
     
  4. hmmm...y must always be +x or -x?
    but i dont understand how the classes are formed. For example class [0], does it mean the list starts at 0? In the b-part the list increases always by 4, but what about in this, by 1?
     
  5. George Jones

    George Jones 6,357
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    Now think about concrete examples. If x = -3, what can y be? Consequently, what is [-3]?
     
  6. if x = -3, then y can then be 3 or -3
    What is [-3]? I dont know, {..., ???, -3, ???, ...}
     
  7. George Jones

    George Jones 6,357
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    Do you understand why the answer to b) is the answer to b)? Back to a).

    [x] = {y | xRy} = {y in Z | |y| = |x|}

    [-3] = {y | (-3)Ry} = {y in Z | |y| = |-3|}
     
  8. HallsofIvy

    HallsofIvy 40,241
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    Yes, if |x|= |y| and x= -3, then y can be either 3 or -3. So the only numbers equivalent to -3 are 3 and -3. [-3] is the set[\b] of all numbers equivalent to -3 so [3]= ?
     
  9. I thought I understood the b) part, but now im not sure if i do deeply enough.

    So, in each class the elements are "equivalent" in the way the equivalence relation is defined? xmod4 = ymod4 means every element which has same modulus when divided by 4 belong to same class?

    can [-3] then be only {-3,3} in the a) -part?
    And [-2] = {-2,2} etc?

    I'm confused because we only had those modulus examples in the class and in book and I dont think I understood the theory deeply enough =)
     
    Last edited: May 3, 2010
  10. George Jones

    George Jones 6,357
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    Yes.
    Yes.
    It looks like you are catching on. :smile:

    What about [0] in a)?
     
  11. [0] must then be only {0}

    What about R: R <-> R, where xRy, iff floor(x) = floor(y)

    i dont know if floor() is the right way to write floor function, but cant find the correct symbol. [2] is then something like {2, 2.1, 2.2, ... , 2.99999...} but what is the correct way to write it? Because 2 can have any amount of decimals after it. Does it have to be in a list form like a) and b) here was?

    Thanks much for the replies, you helped me alot!
     
  12. George Jones

    George Jones 6,357
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    No, it doesn't have to be a list. For example, you can specify [2] by using inequalities.
     
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