4. Determine which properties (reflexive, irreflexive, symmetric, antisymmetric, transitive) does the following relation defined on the set {1, 2, 3, 4} have: R= {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1), (1, 3), (3, 1)}
4. Determine which properties (reflexive, irreflexive, symmetric, antisymmetric, transitive) does the following relation defined on the set {1, 2, 3, 4} have: R= {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1), (1, 3), (3, 1)}
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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Please help with question 4. Thank you.
![**Problem 4: Analyzing the Properties of a Relation**
Determine which properties (reflexive, irreflexive, symmetric, antisymmetric, transitive) the following relation, defined on the set \( \{1, 2, 3, 4\} \), possesses:
\[ R = \{(1, 1), (2, 2), (3, 3), (1, 2), (2, 1), (1, 3), (3, 1)\} \]
**Definitions of Properties:**
- **Reflexive:** A relation \( R \) is reflexive if every element is related to itself. For set \( \{1, 2, 3, 4\} \), this means it must include \( (1, 1), (2, 2), (3, 3), \) and \( (4, 4) \).
- **Irreflexive:** A relation \( R \) is irreflexive if no element is related to itself. For set \( \{1, 2, 3, 4\} \), it must not include \( (1, 1), (2, 2), (3, 3), \) or \( (4, 4) \).
- **Symmetric:** A relation \( R \) is symmetric if for every \( (a, b) \), there is also \( (b, a) \).
- **Antisymmetric:** A relation \( R \) is antisymmetric if, for all \( a \) and \( b \), whenever both \( (a, b) \) and \( (b, a) \) are in \( R \), then \( a = b \).
- **Transitive:** A relation \( R \) is transitive if, whenever \( (a, b) \) and \( (b, c) \) are in \( R \), then \( (a, c) \) is also in \( R \).
Evaluate the given relation \( R \) against these definitions to determine its properties.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F620c60a1-10fe-416e-8b85-1e75a34b227e%2F5c35f4f9-4353-46d2-97ee-68dd14e6c46f%2F88dzal_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem 4: Analyzing the Properties of a Relation**
Determine which properties (reflexive, irreflexive, symmetric, antisymmetric, transitive) the following relation, defined on the set \( \{1, 2, 3, 4\} \), possesses:
\[ R = \{(1, 1), (2, 2), (3, 3), (1, 2), (2, 1), (1, 3), (3, 1)\} \]
**Definitions of Properties:**
- **Reflexive:** A relation \( R \) is reflexive if every element is related to itself. For set \( \{1, 2, 3, 4\} \), this means it must include \( (1, 1), (2, 2), (3, 3), \) and \( (4, 4) \).
- **Irreflexive:** A relation \( R \) is irreflexive if no element is related to itself. For set \( \{1, 2, 3, 4\} \), it must not include \( (1, 1), (2, 2), (3, 3), \) or \( (4, 4) \).
- **Symmetric:** A relation \( R \) is symmetric if for every \( (a, b) \), there is also \( (b, a) \).
- **Antisymmetric:** A relation \( R \) is antisymmetric if, for all \( a \) and \( b \), whenever both \( (a, b) \) and \( (b, a) \) are in \( R \), then \( a = b \).
- **Transitive:** A relation \( R \) is transitive if, whenever \( (a, b) \) and \( (b, c) \) are in \( R \), then \( (a, c) \) is also in \( R \).
Evaluate the given relation \( R \) against these definitions to determine its properties.
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