Define relation R on (Z+ U {0}) × Z+ by (a, b)R(c,d) if ad = bc. Describe the equivalence class [(2, 1)] by listing at least three elements.
Define relation R on (Z+ U {0}) × Z+ by (a, b)R(c,d) if ad = bc. Describe the equivalence class [(2, 1)] by listing at least three elements.
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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![### Relation and Equivalence Class
**Define relation** \( R \) **on** \( (\mathbb{Z}^+ \cup \{0\}) \times \mathbb{Z}^+ \) **by** \( (a, b) R (c, d) \) **if** \( ad = bc \).
**Describe the** *equivalence class* \([ (2,1)]\) **by listing at least three elements**.
### Explanation
In this context, we are defining a relation \( R \) on the set of ordered pairs, where each pair consists of an integer from the set of non-negative integers \((\mathbb{Z}^+ \cup \{0\})\) and a positive integer \((\mathbb{Z}^+)\). The pairs \( (a, b) \) and \( (c, d) \) are related under \( R \) if the product of the first number in the first pair and the second number in the second pair equals the product of the second number in the first pair and the first number in the second pair. Mathematically, this is represented as \( ad = bc \).
To describe the equivalence class \([(2, 1)]\), we need to find at least three pairs \((a, b)\) such that \( (2, 1) R (a, b) \).
Here are three elements that belong to the equivalence class \([ (2,1)]\):
- \((2, 1)\)
- \((4, 2)\)
- \((6, 3)\)
These examples illustrate pairs that satisfy the relation \( ad = bc \):
1. For \((2,1) R (2,1)\): \(2 \cdot 1 = 1 \cdot 2\)
2. For \((2,1) R (4, 2)\): \(2 \cdot 2 = 1 \cdot 4\)
3. For \((2,1) R (6, 3)\): \(2 \cdot 3 = 1 \cdot 6\)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F46272ace-8a35-41ec-9b9a-0375bd37bb55%2F0e9aadde-7a80-4c91-a4cf-9a6f4a1be7f4%2Fz3bfl3e_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Relation and Equivalence Class
**Define relation** \( R \) **on** \( (\mathbb{Z}^+ \cup \{0\}) \times \mathbb{Z}^+ \) **by** \( (a, b) R (c, d) \) **if** \( ad = bc \).
**Describe the** *equivalence class* \([ (2,1)]\) **by listing at least three elements**.
### Explanation
In this context, we are defining a relation \( R \) on the set of ordered pairs, where each pair consists of an integer from the set of non-negative integers \((\mathbb{Z}^+ \cup \{0\})\) and a positive integer \((\mathbb{Z}^+)\). The pairs \( (a, b) \) and \( (c, d) \) are related under \( R \) if the product of the first number in the first pair and the second number in the second pair equals the product of the second number in the first pair and the first number in the second pair. Mathematically, this is represented as \( ad = bc \).
To describe the equivalence class \([(2, 1)]\), we need to find at least three pairs \((a, b)\) such that \( (2, 1) R (a, b) \).
Here are three elements that belong to the equivalence class \([ (2,1)]\):
- \((2, 1)\)
- \((4, 2)\)
- \((6, 3)\)
These examples illustrate pairs that satisfy the relation \( ad = bc \):
1. For \((2,1) R (2,1)\): \(2 \cdot 1 = 1 \cdot 2\)
2. For \((2,1) R (4, 2)\): \(2 \cdot 2 = 1 \cdot 4\)
3. For \((2,1) R (6, 3)\): \(2 \cdot 3 = 1 \cdot 6\)
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