Question

The drawing below shows a Hasse diagram for a partial order on the set {A, B, C, D, E, F, G, H, I, J} D G H E Figure 3: A Hasse diagram shows 10 vertices and 8 edges. The vertices, rep- resented by dots, are as follows: vertex J; vertices H and I are aligned vertically to the right of vertex J; vertices A, B, C, D, and E forms a closed loop, which is to the right of vertices H and I; verter G is inclined upward to the right of verter E; and verter F is inclined downward to the right of verter E. The edges, represented by line segments, between the vertices are as follows: Verter ) is connected to no verter; a vertical edge connects vertices H and I; a vertical edge connects vertices B and C; and 6 inclined edges connect the following vertices, A and B, C and D, D and E, A and E, E and G, and E and F. (a) What are the minimal elements of the partial order? (b) What are the maximal elements of the partial order? (c) Which of the following pairs are comparable? (A, D), (J, F), (B, E),(G, F), (D, B), (C, F), (H, 1), (C, E) Part 3. Each relation given below is a partial order. Draw the Hasse diagram for the partial order. (a) The domain is {3, 5, 6, 7, 10, 14, 20, 30, 60). 15 y if I evenly divides y.

Answer 1

An element is minimal (resp. maximal) if the only
T
with
m
(resp.
C m
) is
T
.

(a) From the diagram we see that the only elements which satisfy the minimality condition are J, I, A and F.

(b) From the diagram we see that the only elements which satisfy the maximality condition are J, H, D and G.

(c) Remember that a partial order is transitive. So, we only need to check if there is a vertical path between the nodes. Therefore,

(A, D) is comparable.
(J, F) is not comparable.
(B, E) is not comparable.
(G, F) is comparable.
(D, B) is comparable.
(C, F) is not comparable.
(H, I) is comparable.
(C, E) is not comparable.

Part 3

(a)

60 30 20 14 - 6 10 7 3 5

(b)

나 으 e d b C