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Math Help - Construction of pairwise disjoint sets

  1. #1
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    [UPDATED PROBLEM] Construction of pairwise disjoint sets

    I am given a problem asking me to identify pairwise disjoint sets that satisfy some property, unfortunately I do not know how I would construct them.

    I'm given sets Y1, Y2, ... and I'm asked to identify pairwise disjoint sets Z1, Z2, ... that satisfy the following property.

    \bigcup_{i=1}^{N}Y_i = \bigcup_{i=1}^{N}Z_i

    I know that if Z1, Z2, ... are pairwise disjoint sets, then \bigcup_{i=1}^{N}Z_i=\sum_{i=1}^{N}Z_i

    Substituting this into the property we get:

    \sum_{i=1}^{N}Z_i= \bigcup_{i=1}^{N}Y_i

    What should be the next step?

    How does this revelation help construct the pairwise disjoint sets and how does it relate to the property?

    EDIT: Rephrased question
    Last edited by chaosier; January 13th 2014 at 02:12 PM.
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  2. #2
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    Re: Construction of pairwise disjoint sets

    Quote Originally Posted by chaosier View Post
    I am given a problem asking me to identify pairwise disjoint sets that satisfy some property, unfortunately I do not know how I would construct them.
    I'm given sets Y1, Y2, ... and I'm asked to identify pairwise disjoint sets Z1, Z2, ... that satisfy the property (UNION Zi for i = 1 to n) = (UNION Yi for i = 1 to n).

    I know that if Z1, Z2, ... are pairwise disjoint sets, then (UNION Zi for i = 1 to n) = sum{from i = 1 to n}(Zi).

    sum{from i = 1 to n}(Zi) = (UNION Yi for i = 1 to n).
    What does sum{from i = 1 to n}(Zi) that mean?

    Having work with set theory for years, I have never seem the expression sum{from i = 1 to n}(Zi)
    In fact, I find the way you asked the question very confusing. I would ask you to repost the questions with a clear explication of the terms.
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    Re: Construction of pairwise disjoint sets

    Sorry, I made up this notation since I don't know how to type with symbols on this forum.

    sum{from i = 1 to N}(Zi) means the summation from i = 1 to N of Z_i, so Z1 + Z2 + Z3 + ... + ZN.
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    Re: Construction of pairwise disjoint sets

    Quote Originally Posted by chaosier View Post
    Sorry, I made up this notation since I don't know how to type with symbols on this forum.

    sum{from i = 1 to N}(Zi) means the summation from i = 1 to N of Z_i, so Z1 + Z2 + Z3 + ... + ZN.
    The point is if each of Z_j is a set then \sum_{i=1}^{N}Z_i has absolutely no meaning whatsoever.

    The summation sign, \sum is not usually used with sets. If you have a textbook that uses it, the you must tell us what it means.
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    Re: Construction of pairwise disjoint sets

    Ok, I suppose that I have the wrong idea then. How would you propose I approach the problem? Looking at my book it seems that it's applied to events, and not sets.
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    Re: Construction of pairwise disjoint sets

    Quote Originally Posted by chaosier View Post
    Ok, I suppose that I have the wrong idea then. How would you propose I approach the problem? Looking at my book it seems that it's applied to events, and not sets.
    I have tried to get you to simply state the problem exactly as it is written.
    Is this part of a probability course?
    Also note there is a difference is a disjoint collection and a pairwise disjoint collection. That distinction is really important.

    Also any collection of sets can be written as pairwise disjoint collection with the same union.
    But if the set are really events then that may be what you want.
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    Re: Construction of pairwise disjoint sets

    Actually the problem is written as I have posted. It tells me to consider sets Y1, Y2, .... It then asks me to identify pairwise disjoint sets Z1, Z2, ... such that the property
    I posted above is satisfied for every N that is an element of {1, 2, 3, ..., }
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    Re: Construction of pairwise disjoint sets

    Quote Originally Posted by chaosier View Post
    I'm given sets Y1, Y2, ... and I'm asked to identify pairwise disjoint sets Z1, Z2, ... that satisfy the following property.
    \bigcup_{i=1}^{N}Y_i = \bigcup_{i=1}^{N}Z_i
    If that is the real question, then let Y=\bigcup_{i=1}^{N}Y_i
    Define Z_1=Y_1 and for 1<J\le N define Z_J=Y_J\setminus \left[\bigcup_{i=1}^{J-1}Z_i\right] .

    I will let you show that works.
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  9. #9
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    Re: Construction of pairwise disjoint sets

    Thanks, I arrived at the same result after much struggling.
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