Given any partially-ordered set  (S,⪯)  , it is known that a standard Hasse diagram can be constructed representing the relation  ⪯   on the vertex set with vertices labelled by the elements of  S  . Gabriel wants to compare the Hasse diagram for a partially-ordered set with the standard arrow diagram for the relation  ⪯   on  S  , which we might refer to as the unsimplified diagram for  (S,⪯)  .     (a)  Consider the set  S={1,2,4,5,6,8,10,12,18}  .  How many edges are in the standard Hasse diagram for the partially-ordered set  (S,∣)  and How many edges are in the unsimplified diagram for  (S,∣)  ? Include the Hasse diagram in your answer.

Here is the Hasse diagram for the set \( S = \{1, 2, 4, 5, 6, 8, 10, 12, 18\} \) with the partial order defined by divisibility:

- **Number of edges in the Hasse diagram**: **11**
- **Number of edges in the unsimplified diagram**: **17**

Do you have any questions or would you like more details?

### Relative Questions:
1. How does the Hasse diagram change if we remove or add an element to the set \( S \)?
2. What is the significance of transitive reduction in the context of Hasse diagrams?
3. How would the Hasse diagram differ if the relation was defined differently, such as a less-than-or-equal relation?
4. What are some common applications of Hasse diagrams in mathematics and computer science?
5. How does the complexity of constructing a Hasse diagram scale with larger sets?

**Tip:** When constructing Hasse diagrams, always check for transitive relations to minimize the number of edges.

Explore the comparison between Hasse diagrams and unsimplified arrow diagrams in partially-ordered sets. Includes examples with set S={1, 2, 4, 5, 6, 8, 10, 12, 18}, detailing the number of edges in the Hasse diagram (11) and unsimplified diagram (17). Discover the significance of transitive relations and their impact on diagram complexity.

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