(1) Draw the Haas diagram of the poset \ Lelang A, R \rangle.
Given the set a = \ {1, 2, 3, 4, 6, 8, 9, 12 \} and the divisible relation r, we need to draw a Haas diagram representing this poset. Divisible relation means that if A divides B equally, then A will be above B in Haas diagram, and there will be a line segment between them.
First, we find the factor of each element in the set:
1 is divisible by all elements.
2 divided by 4,6,8, 12.
3 is divisible by 12.
4 is divisible by 12.
6 is divisible by 12.
8 is divisible by 12.
12 is the largest element, and there is no factor in the set.
Based on this information, we can draw a Hastings diagram. Because this diagram is complicated, I will describe its structure in words:
1 At the bottom, connect all elements.
2 is located above 1 and connected with 4, 6, 8, 12.
3 is above 1 and only connected to 12.
4 is above 2, only connected to 12.
6 is above 2 and 3, only connected to 12.
8 is above 2, only connected to 12.
12 is at the top and is not connected with other elements.
(2) Find the element attribute of subset b = \ {2,4,6, 12\}
For subset b, we need to find out the attributes of the following elements:
Maximum element: the largest element in a set. This is 12.
Minimum element: the smallest element in a set. This is 2 pounds.
The largest element: an element that is not the predecessor of other elements. Here, 12 is the largest element because it has no successor elements.
Minimum element: an element without a predecessor. Here, 2 is the smallest element because it has no predecessor.
Upper bound: All elements in the set are less than or equal to its elements. 12 is an upper bound of b, but it is not unique, because any element greater than or equal to 12 is an upper bound of b.
Supremum: the smallest element of the upper bound of a set. 12 is the supremum of b.
Lower bound: an element that is less than or equal to all elements in the set. 2 is the lower bound of B, but it is not unique, because any element less than 2 is the lower bound of B.
Lower supremum: The largest element in the lower bound of a set. 2 is the lower bound of B.
Please note that since B is a subset selected from A, the definition of upper and lower bounds is relative to B itself, not to the whole set of real numbers or larger sets. In this particular problem, 12, as the largest element of B, is naturally the upper bound and supremum of B; As the smallest element of B, 2 is also the lower bound and supremum of B.