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Describe the partition for the equivalence relation T

  1. Jun 30, 2008 #1
    For the set A = {1,2,3,4,5,6,7}, determine whether script A is a partition of A. script A = {{1,3,},{5,6}, {2,4},{7}}

    Describe the partition for the equivalence relation T defined for x,y [tex] \in \mathbbc{R} [/tex] by X T y iff [tex] \left[ \left[x \right] \right] = \left[ \left[y \right] \right] where \left[ \left[x \right] \right] [/tex] is definied to be the greatest integer iin x (the largest integer n such that n [tex] \leq [/tex] x).

    Can anyone help me with this partition stuff. It would be very appreciated. :)
  2. jcsd
  3. Jun 30, 2008 #2


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    Re: Partitions

    Can you start by giving the definition of a partition?
    Then try to check if the given sets [itex]A, \mathcal{A}[/itex] satisfy this definition.

    For the second one, can you imagine what the equivalence classes look like?
  4. Jun 30, 2008 #3
    Re: Partitions

    (i) If X [tex] \in \mathcal{A}, [/tex] then X [tex] \neq \o [/tex]
    (ii) If X [tex] \in \mathcal{A} [/tex] and Y [tex]\in \mathcal{A}[/tex], then X=Y or X [tex]\cap [/tex] Y= [tex]\o[/tex]
    (iii)[tex]\bigcup_{X \in \mathal{A}}X=A[/tex]

    I found this to be a definition for an equivalence class:
    In mathematics, given a set X and an equivalence relation ~ on X, the equivalence class of an element a in X is the subset of all elements in X which are equivalent to a:

    [tex]\left[a\right]={x \in X|x \sim a} [/tex]
  5. Jun 30, 2008 #4


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    Re: Partitions

    That's long winded! A "partition" of a set, A, is a collection of subsets of A such that every member of A is in one and only one of the subsets. Here, the members of A are 1, 2, 3, 4, 5, 6, 7. Is every one of those numbers in one of the given subsets? Is any number in more than one?

    Okay, and the relation x T y is defined by "x T y if and only if the largest integer less than or equal to x is the same as the largest integer less than or equal to y".

    Now try some examples. What numbers are equivalent to 0? to 1/2? to 5/4? to [itex]\pi[/itex]?
  6. Jun 30, 2008 #5
    Re: Partitions

    All the numbers in a subset are only in one subset once.

    -What do you mean by what numbers are equivalent to 0, 1/2, pi, etc
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