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19 May, 13:48

How many distinct pairs of disjoint non-empty subsets of A are there, the union of which is all of A? A {0,4,5,7}

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  1. 19 May, 14:04
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    Let set A be written as B∪C, where B and C are 2 disjoint nonempty subsets of A.

    the number of elements of B and C respectively can be:

    i) 1, 3

    ii) 2, 2

    for case i we have the following pairs of sets:

    {0}, {4,5,7}

    {4}, {0,5,7}

    {5}, {0,4,5}

    {7}, {0,4,5}

    for case ii, consider only one of the sets, which can be any of these:

    {0,4}, {0,5}, {0,7}, {4,5}, {4,7},{5,7}

    Clearly these pairs complement each other, that is if the first set is {0,4}, the second set is {5,7}, if the first set is {0,5} the second set is {4,7}, and if the first set is {0,7} the second is {4,5}.

    Thus there are 4+3=7 pairs of subsets

    Answer: 7
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