Math Problem Statement
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.
Solution
Here is the Hasse diagram for the set 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:
- How does the Hasse diagram change if we remove or add an element to the set ?
- What is the significance of transitive reduction in the context of Hasse diagrams?
- How would the Hasse diagram differ if the relation was defined differently, such as a less-than-or-equal relation?
- What are some common applications of Hasse diagrams in mathematics and computer science?
- 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.
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Math Problem Analysis
Mathematical Concepts
Partially-ordered sets
Hasse diagrams
Divisibility relation
Formulas
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Theorems
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Suitable Grade Level
Advanced Mathematics