Draw the Hasse diagram for following set.
(𝑎) {(𝑎, 𝑎), (𝑎, 𝑏), (𝑎, 𝑐), (𝑎, 𝑑), (𝑏, 𝑏), (𝑏, 𝑐), (𝑏, 𝑑), (𝑐, 𝑐), (𝑑, 𝑑)}

To draw the **Hasse diagram** for the given set of ordered pairs, let's first analyze the given relations. We are provided with the following set of relations:

\[
\{(a, a), (a, b), (a, c), (a, d), (b, b), (b, c), (b, d), (c, c), (d, d)\}
\]

### Step-by-Step Process to Create the Hasse Diagram:

1. **Understand the Partial Order**:
   - The relation appears to be a partial order on the elements \( \{a, b, c, d\} \), where each pair indicates \( x \leq y \).
   - The elements are ordered such that \(a \leq b \leq c \leq d\), because we see:
     - \(a \leq b\), \(a \leq c\), \(a \leq d\)
     - \(b \leq c\), \(b \leq d\)
     - \(c \leq d\)

2. **Remove Reflexive Pairs**:
   - Reflexive pairs, like \( (a, a) \), \( (b, b) \), \( (c, c) \), \( (d, d) \), are not drawn in the Hasse diagram, since they are implied for every element in the partial order.

3. **Remove Transitive Pairs**:
   - In a Hasse diagram, transitive relations are not drawn. For example:
     - \(a \leq c\) can be inferred from \(a \leq b\) and \(b \leq c\), so we remove \( (a, c) \).
     - Similarly, \(a \leq d\) can be inferred via \(a \leq b\) and \(b \leq d\), so we remove \( (a, d) \).
     - Also, \(b \leq d\) can be inferred from \(b \leq c\) and \(c \leq d\), so we remove \( (b, d) \).

4. **Remaining Pairs for the Hasse Diagram**:
   - After removing reflexive and transitive pairs, the remaining pairs are:
     \[
     \{(a, b), (b, c), (c, d)\}
     \]

### Hasse Diagram:
The Hasse diagram is a simple chain where \(a\) is at the bottom and \(d\) is at the top, and the elements are connected in a linear fashion:

```
d
|
c
|
b
|
a
```

This diagram shows that:
- \(a \leq b\),
- \(b \leq c\),
- \(c \leq d\).

Would you like further clarification or details on how the diagram was constructed?

### Related Questions:
1. What are reflexive, antisymmetric, and transitive properties in partial orders?
2. How does a Hasse diagram represent partial orders more clearly than a general diagram?
3. What is the difference between a total order and a partial order?
4. Can a Hasse diagram have cycles, or must it always be acyclic?
5. How can the Hasse diagram be used to find the maximal and minimal elements of a set?

### Tip:
Always remove reflexive and transitive pairs when constructing a Hasse diagram to simplify the representation.

Draw the Hasse diagram for the following set: {(a, a), (a, b), (a, c), (a, d), (b, b), (b, c), (b, d), (c, c), (d, d)}

Learn how to draw the Hasse diagram for a set with partial orders, covering reflexivity, antisymmetry, and transitivity. The solution simplifies the set {(a, a), (a, b), (a, c), (a, d), (b, b), (b, c), (b, d), (c, c), (d, d)} to show relations between elements. Suitable for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation