Thread: 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.

First I think that you have notational problems.
You have .
I think that you want: .

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.