Explain whny the following binary relation on. 1,2,3} Is not R= {(1,1)(1,2),(1,3),(2,1).(2,2).(3,3)}|
Explain whny the following binary relation on. 1,2,3} Is not R= {(1,1)(1,2),(1,3),(2,1).(2,2).(3,3)}|
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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![### Relations and Equivalence Relations
#### Question:
**2. Explain why the following binary relation on S = {1, 2, 3} is not an equivalence relation on S.**
The relation \( R \) is defined as follows:
\[ R = \{ (1,1), (1,2), (1,3), (2,1), (2,2), (3,3) \} \]
#### Solution:
To determine why \( R \) is not an equivalence relation on the set \( S = \{1, 2, 3\} \), we must evaluate it with respect to the three properties of equivalence relations:
1. **Reflexivity**:
- For a relation to be reflexive, every element must be related to itself. Specifically, for all \( x \in S \), the pair \( (x, x) \) must be in \( R \).
- In this case, the pairs \( (1,1) \), \( (2,2) \), and \( (3,3) \) are present in \( R \), so the relation is reflexive.
2. **Symmetry**:
- For a relation to be symmetric, if \( (a,b) \in R \), then \( (b,a) \) must also be in \( R \).
- Here, we see that both \( (1,2) \) and \( (2,1) \) are present, which satisfies the symmetric property for these two pairs. However, \( (1,3) \) is in \( R \) but \( (3,1) \) is not in \( R \). This violates the symmetry requirement.
3. **Transitivity**:
- For a relation to be transitive, if \( (a,b) \in R \) and \( (b,c) \in R \), then \( (a,c) \) must also be in \( R \).
- In this set, for example, \( (1, 2) \in R \) and \( (2,1) \in R \) imply that \( (1,1) \in R \), which is true. However, \( (1,2) \in R \) and \( (2,3) \in R \) should imply \( (1,3)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8819457a-80f1-4499-912b-dbec38bab78a%2Ff6a79fdd-6b85-4fe6-b595-bdfff134f009%2Fffag4qf.jpeg&w=3840&q=75)
Transcribed Image Text:### Relations and Equivalence Relations
#### Question:
**2. Explain why the following binary relation on S = {1, 2, 3} is not an equivalence relation on S.**
The relation \( R \) is defined as follows:
\[ R = \{ (1,1), (1,2), (1,3), (2,1), (2,2), (3,3) \} \]
#### Solution:
To determine why \( R \) is not an equivalence relation on the set \( S = \{1, 2, 3\} \), we must evaluate it with respect to the three properties of equivalence relations:
1. **Reflexivity**:
- For a relation to be reflexive, every element must be related to itself. Specifically, for all \( x \in S \), the pair \( (x, x) \) must be in \( R \).
- In this case, the pairs \( (1,1) \), \( (2,2) \), and \( (3,3) \) are present in \( R \), so the relation is reflexive.
2. **Symmetry**:
- For a relation to be symmetric, if \( (a,b) \in R \), then \( (b,a) \) must also be in \( R \).
- Here, we see that both \( (1,2) \) and \( (2,1) \) are present, which satisfies the symmetric property for these two pairs. However, \( (1,3) \) is in \( R \) but \( (3,1) \) is not in \( R \). This violates the symmetry requirement.
3. **Transitivity**:
- For a relation to be transitive, if \( (a,b) \in R \) and \( (b,c) \in R \), then \( (a,c) \) must also be in \( R \).
- In this set, for example, \( (1, 2) \in R \) and \( (2,1) \in R \) imply that \( (1,1) \in R \), which is true. However, \( (1,2) \in R \) and \( (2,3) \in R \) should imply \( (1,3)
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