Consider the “divides” relation on each of the following
sets A. Draw the Hasse diagram for each
relation.
b. A 5 {2, 3, 4, 6, 8, 9, 12, 18}

To create the Hasse diagram for the "divides" relation on the set \( A = \{2, 3, 4, 6, 8, 9, 12, 18\} \), we need to consider the partial order induced by the divides relation. In this relation, \( a \leq b \) if and only if \( a \) divides \( b \) (written as \( a \mid b \)).

### Steps to Build the Hasse Diagram

1. **Identify All Relations**:
   For each pair \( (a, b) \) in \( A \), determine if \( a \mid b \).
   
2. **Remove Transitive Edges**:
   In a Hasse diagram, we exclude edges that can be inferred by transitivity to simplify the structure. For example, if \( a \mid b \) and \( b \mid c \), then we exclude \( a \mid c \) directly.

3. **Draw the Diagram**:
   Place each element of \( A \) as a vertex. Connect vertices with lines only if one element directly divides another without any intermediary.

### Analysis of Relations

1. **Starting with Smallest Elements**:
   - \( 2 \): Divides \( 4, 6, 8, 12, \) and \( 18 \).
   - \( 3 \): Divides \( 6, 9, \) and \( 18 \).
   
2. **Building on Larger Elements**:
   - \( 4 \): Divides \( 8 \) and \( 12 \).
   - \( 6 \): Divides \( 12 \) and \( 18 \).
   - \( 8 \): Does not divide any larger elements within \( A \).
   - \( 9 \): Divides \( 18 \).
   - \( 12 \) and \( 18 \): Are the largest elements in \( A \) for this relation.

### Resulting Hasse Diagram

The Hasse diagram will have the following structure:

- **Bottom Layer**: \( 2 \) and \( 3 \), as they divide other numbers.
- **Middle Layers**:
   - **First Level from Bottom**: \( 4 \), \( 6 \), \( 8 \), and \( 9 \).
   - **Second Level**: \( 12 \) and \( 18 \).
- **Top Layer**: None, as \( 18 \) and \( 12 \) are the largest in the set.

Here’s the basic structure for the Hasse Diagram based on the given relation:

```
           18
          /   \
         6     9
        / \    |
       2   4   3
      / \   \ / 
     12    8  
```

Would you like a more detailed, step-by-step explanation or a specific layout in graph form?

### Related Questions

1. How would the Hasse diagram change if we included 1 in the set?
2. What is the significance of transitive relations in a Hasse diagram?
3. Can we extend this concept to other types of divisibility relations, such as modulo relationships?
4. How would the diagram differ if we considered a different partial order, such as "greater than"?
5. Why is the divides relation useful in understanding number structure?

#### Tip:
Always remove transitive edges in Hasse diagrams to simplify and focus on direct relationships.

Consider the “divides” relation on each of the following sets A. Draw the Hasse diagram for each relation. b. A = {2, 3, 4, 6, 8, 9, 12, 18}

Explore the construction of the Hasse diagram for the divides relation on set A = {2, 3, 4, 6, 8, 9, 12, 18}. This example demonstrates key concepts in partial order and set theory, focusing on how to create and simplify Hasse diagrams for direct relationships without transitive edges.

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