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Math Help - Partitioning of Sets

  1. #1
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    Partitioning of Sets

    Hey guys. I have this problem:

    Let A = \left\{1,2,...,12\right\}. Give an example of a partition S of A satisfying the following requirements:
    (i) |S| = 5,
    (ii) there is a subset T of S such that |T| = 4 and |\cup_{X\in T} X| = 10 and
    (iii) there is no element B \in S such that |B| = 3.


    I need help understanding what the second part of (ii) means. I was able to create a partition that I think satisfies the other conditions. I just don't see how |\cup_{X\in T} X| could have a cardinality of 10 when the subset T has only 4 elements.

    S=\left\{\left\{1,2,3,4\right\},\left\{5,6\right\}  ,\left\{7,8\right\},\left\{9,10\right\},\left\{11,  12\right\} \right\}

    subset T=\left\{1,2,3,4\right\}
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  2. #2
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    Re: Partitioning of Sets

    Hi,
    What are the elements of a partition S? Why, they are subsets of A. So if T is to be a subset of S, elements of T are subsets of A. Your T does not consist of subsets of A; e.g. 1 is an element of T, but 1 is certainly not an element of S.

    You've almost solved your problem. Take T to be a 4 element subset of S with the union of the subsets of A in T totaling 10 integers.
    Thanks from amthomasjr
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  3. #3
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    Re: Partitioning of Sets

    Quote Originally Posted by johng View Post
    Hi,
    What are the elements of a partition S? Why, they are subsets of A. So if T is to be a subset of S, elements of T are subsets of A. Your T does not consist of subsets of A; e.g. 1 is an element of T, but 1 is certainly not an element of S.

    You've almost solved your problem. Take T to be a 4 element subset of S with the union of the subsets of A in T totaling 10 integers.
    I'm understanding it little by little. The elements of a partition S are \left\{1,2,3,4\right\},\left\{5,6\right\}, \left\{7,8\right\}, \left\{9,10\right\}, \left\{11,12\right\}
    They are subsets because the elements of the subsets are in A but not every element of A are in the subsets.
    So I had T as subsets of A instead of S. So I could have:

    T=\left\{\left\{1,2,3,4\right\},\left\{5,6\right\}  , \left\{7,8\right\}, \left\{9,10\right\}\right\} making the cardinality 4.

    So \bigcup_{X\in T}X=\left\{1,2,3,4,5,6,7,8,9,10\right\} making the cardinality of the union of the subsets equal to 10, and it now satisfies all of the conditions.
    Last edited by amthomasjr; August 29th 2013 at 08:17 PM.
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