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.