DHARM
154 MATHEMATICAL FOUNDATION OF COMPUTER SCIENCE
Hence the Fig. 6.5 shows the Hasse diagram for poset (S, ≤)
24
12
18
9
9
4
8
1
3
2
Fig. 6.5
Example 6.41. Draw the Hasse diagram for factors of 6 under relation divisibility.
- Draw the Hasse diagram for factors of 8 under relation divisibility.
Sol. 1. Let D 6 is the set that contains possible elements that are factors of 6, i.e., D 6 = {1,
2, 3, 6} then poset will be {{1, 2}, {1, 3}, {2, 6}, {3, 6} whose Hasse diagram is shown in Fig. 6.6.
6
23
1
Fig. 6.6
- Similarly for D 8 = {1, 2, 4, 8} under the relation divisibility the poset will be {1, 2}, {2, 4},
{4, 8}. Hence, its Hasse diagram will be a chain that is shown in Fig. 6.7. Here all elements are
comparable to each other such poset is called toset. Remember, all tosets must be posets but
all posets are not necessarily tosets.
8
4
2
1
Fig. 6.7
Upper Bound
Let (X, ≤) be a poset and Y ⊆ X, then an element x ∈ X be the upper bound for Y if and only if,
∀y ∈ Y s.t. y ≤ x.
