
7.	Show Draw the Hasse diagram for the divisibility on the set { 1, 2, 3, 6, 12, 24, 36, 48, 96 } and hence find the maximal and minimal members.

To solve this problem, let's break it down into two main tasks:

1. **Draw the Hasse Diagram**: The Hasse diagram is a graphical representation of a partially ordered set (poset). In this case, the poset is the set \(\{1, 2, 3, 6, 12, 24, 36, 48, 96\}\) ordered by divisibility. In a Hasse diagram, each element is represented as a node, and there's an edge between two nodes \(a\) and \(b\) if \(a\) divides \(b\) and there's no element \(c\) such that \(a\) divides \(c\) and \(c\) divides \(b\) (i.e., \(c\) lies between \(a\) and \(b\)).

2. **Find the Maximal and Minimal Members**: The maximal members of the set are those that are not divisible by any other element in the set, while the minimal members are those that do not divide any other element in the set (except themselves).

### Step 1: Construct the Hasse Diagram
Here’s how we construct the Hasse diagram:

1. **List the set elements**: {1, 2, 3, 6, 12, 24, 36, 48, 96}.
2. **Determine divisibility relations**: 
   - 1 divides everything.
   - 2 divides 6, 12, 24, 48, and 96.
   - 3 divides 6, 12, 36, and 96.
   - 6 divides 12, 24, 36, and 96.
   - 12 divides 24, 36, and 96.
   - 24 divides 48 and 96.
   - 36 divides 96.
   - 48 divides 96.

3. **Draw the diagram**: The elements are arranged vertically. Start with 1 at the bottom, as it is divisible by every other element. Then, connect each element to those it divides directly without passing through an intermediate element.

### Step 2: Identify Maximal and Minimal Members
- **Minimal members**: The minimal elements in this set are those that have no elements below them except 1. This is simply the smallest number, which is 1.
- **Maximal members**: The maximal elements are those that have no elements above them. In this case, it's 96, as no other element divides it.

### Hasse Diagram

The Hasse diagram based on the above steps would look like this:

```
    96
   /  |  \
 48   36  24
  |   /  / 
  12    6
  | \   |
  2   3
  \  / 
   1
```

### Maximal and Minimal Members
- **Maximal member**: \(96\)
- **Minimal member**: \(1\)

Would you like any more details or have any other questions? Here are some related questions you might find interesting:

1. How do we construct Hasse diagrams for larger sets?
2. What other methods can be used to visualize partial orders?
3. How do minimal and maximal elements relate to least upper bounds and greatest lower bounds?
4. Can you have multiple maximal or minimal elements in a Hasse diagram?
5. How does divisibility affect the structure of the Hasse diagram?

**Tip**: When drawing Hasse diagrams, always ensure that you remove any redundant edges to accurately represent the partial order without cluttering the diagram.

Show Draw the Hasse diagram for the divisibility on the set { 1, 2, 3, 6, 12, 24, 36, 48, 96 } and hence find the maximal and minimal members.

Explore how to draw the Hasse diagram for the divisibility on the set {1, 2, 3, 6, 12, 24, 36, 48, 96} and identify the maximal and minimal members. This problem involves key concepts in set theory and partial order relations, making it suitable for students in Grades 9-12.

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