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Math Help - set partitions

  1. #1
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    set partitions

    I've been stuck on this question for a while now and was hoping someone might be able to give me some help cause i don't really understand it.

    Let B be a set. Define what it means to say that
    B1,B2 give a partition of B, where B1 and B2 are subsets of B. Let
    f : A −→ B be a function. Suppose that C is a subset of B. Write down
    the definition of f−1(C). Suppose that B1,B2 is a partition of B. Prove,
    using your definitions, that f−1(B1), f−1(B2) is a partition of A
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  2. #2
    Senior Member Dinkydoe's Avatar
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    B_1,B_2 is a partition of B when:

    1. B_1\cap B_2 = \emptyset
    2. B_1,B_2\neq \emptyset
    3. B_1\cup B_2 = B

    Denote B = B_1\oplus B_2 as the disjunct union of B_1,B_2.

    If f:A\to B is a function and C\subset B. You can write C = (C\cap B_1)\oplus(C\cap B_2). Then f^{-1}(C) = f^{-1}(C\cap B_1\oplus C\cap B_2) =  f^{-1}(C\cap B_1)\oplus f^{-1}(C\cap B_2) = P_1\oplus P_2.
    Then we've shown that P_1,P_2 form a partition of f^{-1}(C)\subset A.

    The same way is shown that f^{-1}(B) = f^{-1}(B_1)\oplus f^{-1}(B_2) = P_1\oplus P_2 = A.

    You simply need to show that P_1,P_2 satisfy conditions 1,2,3
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  3. #3
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    Let B be a set. Define what it means to say that
    B1,B2 give a partition of B, where B1 and B2 are subsets of B.
    If you have a collection of nonempty subsets of B which are pairwise disjoint (any two distinct ones are disjoint), and the union of all these subsets is the set itself, then you know those subsets are partition of that set.

    If B_1 and B_2 are (1) nonempty, (2) non-overlapping subsets and (3) B_1\cup B_2 = B, you know they are partitions of of the set B. These 3 conditions are what partition means; for example gender partitions humanity.
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