Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
expand_more
expand_more
format_list_bulleted
Related questions
Question
![### 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)](https://content.bartleby.com/qna-images/question/8819457a-80f1-4499-912b-dbec38bab78a/f6a79fdd-6b85-4fe6-b595-bdfff134f009/ffag4qf.jpeg)
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)
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by stepSolved in 2 steps with 2 images
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.Similar questions
- Let ? = {1, 2, 3, 4, 5, 6, 7} and define a relation ? ⊆ ? × ? by? = {(1,1), (1,2), (1,3), (2,3), (7,1), (7,2), (7,3), (4,5), (4,6), (5,6), (5,5)}Determine (yes or no) whether ? has the following properties (no justification is required).arrow_forwardLet A={1,2,3,4} and let R be the relation from A to A R={(1,1),(1,4),(2,1),(2,2),(2,3),(3,1),(3,4),(4,3)} Create thr diagraph and matrix Mr for the relationarrow_forwarddiscrete matharrow_forward
Recommended textbooks for you
- Advanced Engineering MathematicsAdvanced MathISBN:9780470458365Author:Erwin KreyszigPublisher:Wiley, John & Sons, Incorporated
Numerical Methods for EngineersAdvanced MathISBN:9780073397924Author:Steven C. Chapra Dr., Raymond P. CanalePublisher:McGraw-Hill Education
Introductory Mathematics for Engineering Applicat...Advanced MathISBN:9781118141809Author:Nathan KlingbeilPublisher:WILEY 
Mathematics For Machine TechnologyAdvanced MathISBN:9781337798310Author:Peterson, John.Publisher:Cengage Learning,
Advanced Engineering Mathematics
Advanced Math
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:9780073397924
Author:Steven C. Chapra Dr., Raymond P. Canale
Publisher:McGraw-Hill Education
Introductory Mathematics for Engineering Applicat...
Advanced Math
ISBN:9781118141809
Author:Nathan Klingbeil
Publisher:WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:9781337798310
Author:Peterson, John.
Publisher:Cengage Learning,