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Math Help - supremum and infimum of a set

  1. #1
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    supremum and infimum of a set

    M={1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}
    A={3, 4, 5}, B={9, 12}, C={1, 4, 8}.
    The graph is attached.

    I need to find the supremum, infimum, max, min of the sets, the biggest and the smallest element.
    I have the following answer in the book.

    М - hasn't got a smallest and biggest element
    min elements are 11 и 12.
    max elements are 6, 7, 9 и 10.
    А*={7}, supA=7, A*={1, 11}, infA=1
    B*=∅, supB hasn't got , B*=∅, infB hasn't got
    C*={8, 10}, supC=8, C*={1, 11}, infC=1.

    I have no idea how they got he answers. I hope you can help me with this one.
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  2. #2
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    First I think that you have notational problems.
    You have \left\{ 7 \right\} = A^*  = \left\{ {1,11} \right\}.
    I think that you want: \left\{ 7 \right\} = A^* \,\& \,A_*  = \left\{ {1,11} \right\}.

    From the diagram 11 precedes every element except 12. In order to have a smallest element it must precede every element. Likewise, there is no element that is preceded by every element, so there is no greatest element.

    Now consider the set A. Both 1 & 11 precede every element of A. Therefore, each of 1 & 11 is a lower bound of A. But 1 is the inf(A) because it is the greatest lower bound.

    Now consider the set C. Both 8 & 10 are preceded by each element of C (and yes 8 is said to precedes itself). But 8 is the sup(C) because it is the least upper bound.
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