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Set theory proof 2

  1. Apr 2, 2016 #1
    1. The problem statement, all variables and given/known data
    . Let A be a set and {B1, B2, B3} a partition of A. Assume {C11, C12} is a partition of B1, {C21, C22} is a partition of B2 and {C31, C32} is a partition of B3. Prove that {C11, C12, C21, C22, C31, C32} is a partition of A.

    2. Relevant equations


    3. The attempt at a solution
    I know this problem looks a little bit intuitive but I just want to make sure I make sense
    proof

    so {C11, C12, C21, C22, C31, C32} is a partition of A because A is divided into 3 disjoint subsets which make a partition of A, B1,B2,B3. And then each subset of these 3 disjoint subsets is further divided into two disjoint subsets which make a partition of the parent subset. So all elements in the parent subsets will make a partition of A because every element in each parent subset are disjoint among the elements of every other parent subset.
     
  2. jcsd
  3. Apr 2, 2016 #2

    Orodruin

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    Your idea is correct, but could be put in a bit more formal way, i.e., take an element in A and show that it must belong to one and only one of the sets Cij.
     
  4. Apr 2, 2016 #3
    ok this is what I did
    proof
    x∈A
    so x∈(Just one of B1,B2,B3 because disjoint by definition)
    case 1 x∈A
    so x∈ Just one of C11,C12 because disjoint by definition

    case 2 x∈B
    so x∈ Just one of C21,C22 because disjoint by definition

    case 3 x∈C
    so x∈ Just one of C31,C32 because disjoint by definition

    which implies that x∈(Just one of C11,C12,C21,C22,C31,C32)
     
  5. Apr 3, 2016 #4

    Orodruin

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    I would also specify that x being in B1 rules out it being in any other C
     
  6. Apr 3, 2016 #5

    HallsofIvy

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    You mean "x∈B1"

    You mean "x∈B2"

    You mean "x∈B3"

     
    Last edited by a moderator: Apr 3, 2016
  7. Apr 3, 2016 #6
    yes, didn't notice that. Is the logic right?. How does this show that these elements make a partition of A?
     
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