8. Let S = {(a, b), (b, a), (b, d), (b, e), (b, f), (c, c), (d, e), (f,c), (f, f)} be a relation on the set {a, b, c, d, e, f}. Write the relation S as a set of ordered pairs.
8. Let S = {(a, b), (b, a), (b, d), (b, e), (b, f), (c, c), (d, e), (f,c), (f, f)} be a relation on the set {a, b, c, d, e, f}. Write the relation S as a set of ordered pairs.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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I need help closure and composition
number 8
![**8. Problem Statement:**
Let
\[ S = \{ (a, b), (b, a), (b, d), (b, e), (b, f), (c, c), (d, e), (f, c), (f, f) \} \]
be a relation on the set \(\{ a, b, c, d, e, f \}\). Write the relation \( S^2 \) as a set of ordered pairs.
---
**Explanation:**
- **Relation \( S \):** A set of ordered pairs that define how elements are related within a set.
- **Set Elements:** \( \{ a, b, c, d, e, f \} \)
- **Objective:** To find \( S^2 \), we need to compute the composition of \( S \) with itself. This involves finding pairs \((x, z)\) such that there exists an intermediate \( y \) where both \((x, y)\) and \((y, z)\) are in \( S \).
---
**Instructions for Calculating \( S^2 \):**
1. **Identify Intermediate Connections:**
- Find pairs \((x, y)\) in \( S \).
- Check if there exists a \((y, z)\) in \( S \).
2. **Form New Pairs:**
- Based on intermediate connections, create new pairs \((x, z)\).
3. **Compile \( S^2 \):**
- Gather all new pairs to form \( S^2 \).
Ensure to systematically explore all possible intermediate connections to complete the set \( S^2 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F71b4fcba-08e1-4941-b15c-eaa6a4e7d487%2F1bf968dc-6fdf-4088-9ebe-ed457322fc3f%2F7ouc89d_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**8. Problem Statement:**
Let
\[ S = \{ (a, b), (b, a), (b, d), (b, e), (b, f), (c, c), (d, e), (f, c), (f, f) \} \]
be a relation on the set \(\{ a, b, c, d, e, f \}\). Write the relation \( S^2 \) as a set of ordered pairs.
---
**Explanation:**
- **Relation \( S \):** A set of ordered pairs that define how elements are related within a set.
- **Set Elements:** \( \{ a, b, c, d, e, f \} \)
- **Objective:** To find \( S^2 \), we need to compute the composition of \( S \) with itself. This involves finding pairs \((x, z)\) such that there exists an intermediate \( y \) where both \((x, y)\) and \((y, z)\) are in \( S \).
---
**Instructions for Calculating \( S^2 \):**
1. **Identify Intermediate Connections:**
- Find pairs \((x, y)\) in \( S \).
- Check if there exists a \((y, z)\) in \( S \).
2. **Form New Pairs:**
- Based on intermediate connections, create new pairs \((x, z)\).
3. **Compile \( S^2 \):**
- Gather all new pairs to form \( S^2 \).
Ensure to systematically explore all possible intermediate connections to complete the set \( S^2 \).
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