Consider the partial order with domain{3,5,6,7,10,14,20,30,60,70}and with x≤y if x evenly divides y. Select the correct Hasse diagram for the partial order

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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Consider the partial order with domain{3,5,6,7,10,14,20,30,60,70}and with x≤y if x evenly divides y. Select the correct Hasse diagram for the partial order

 

**Figure 2**: A Hasse diagram shows a set of elements 3; 5; 6; 7; 10; 14; 20; 30; 60; 70. The diagram illustrates the partial ordering of these elements. There are lines connecting 3 and 6, 6 and 30, 30 and 60, 5 and 10, 10 and 20, 20 and 60, 10 and 70, 7 and 14, 14 and 70.

### Detailed Explanation:

- **Elements:** 
  - The set comprises the numbers: 3, 5, 6, 7, 10, 14, 20, 30, 60, 70.
  
- **Lines Representing Relations:** 
  - **3 is connected to 6**
  - **6 is connected to 30**
  - **30 is connected to 60**
  - **5 is connected to 10**
  - **10 is connected to 20**
  - **20 is connected to 60**
  - **10 is connected to 70**
  - **7 is connected to 14**
  - **14 is connected to 70**

This Hasse diagram visually represents the hierarchical relationships and order between the elements based on divisibility or other criteria within the poset (partially ordered set). Each line indicates a direct connection without intermediaries, demonstrating the principles of transitivity and reduction in visualization.
Transcribed Image Text:**Figure 2**: A Hasse diagram shows a set of elements 3; 5; 6; 7; 10; 14; 20; 30; 60; 70. The diagram illustrates the partial ordering of these elements. There are lines connecting 3 and 6, 6 and 30, 30 and 60, 5 and 10, 10 and 20, 20 and 60, 10 and 70, 7 and 14, 14 and 70. ### Detailed Explanation: - **Elements:** - The set comprises the numbers: 3, 5, 6, 7, 10, 14, 20, 30, 60, 70. - **Lines Representing Relations:** - **3 is connected to 6** - **6 is connected to 30** - **30 is connected to 60** - **5 is connected to 10** - **10 is connected to 20** - **20 is connected to 60** - **10 is connected to 70** - **7 is connected to 14** - **14 is connected to 70** This Hasse diagram visually represents the hierarchical relationships and order between the elements based on divisibility or other criteria within the poset (partially ordered set). Each line indicates a direct connection without intermediaries, demonstrating the principles of transitivity and reduction in visualization.
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