etermine whether the relation defined below is reflexive, symmetric, and/or transitiv _swer. R is the relation on Z where x Ry if x + y is even.
etermine whether the relation defined below is reflexive, symmetric, and/or transitiv _swer. R is the relation on Z where x Ry if x + y is even.
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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![**Determining Properties of a Given Relation**
**Problem Statement:**
Determine whether the relation defined below is reflexive, symmetric, and/or transitive. Justify your answer.
**Given Relation:**
\[ R \text{ is the relation on } \mathbb{Z} \text{ where } xRy \text{ if } x + y \text{ is even}.\]
*Explanation:*
- **Reflexive**: A relation \(R\) on a set \(A\) is reflexive if every element is related to itself. In other words, for every \(x \in A\), \(xRx\) holds true.
- **Symmetric**: A relation \(R\) on a set \(A\) is symmetric if for every \(x, y \in A\), whenever \(xRy\) holds, \(yRx\) also holds.
- **Transitive**: A relation \(R\) on a set \(A\) is transitive if for every \(x, y, z \in A\), whenever \(xRy\) and \(yRz\) hold, \(xRz\) also holds.
**Justification Requirements:**
- For reflexivity: Verify if \( xRx \) based on the given condition \( x + x = 2x \), and determine if this is always even.
- For symmetry: Check if \( xRy \) implies \( yRx \) based on the given condition \( x + y \).
- For transitivity: Determine if \( xRy \) and \( yRz \) imply \( xRz \) based on the sum properties \( x + y \) and \( y + z \).
This problem involves understanding the mathematical definitions of reflexive, symmetric, and transitive relations, and applying them to the given condition related to even sums.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7011b814-0888-4e7c-ad26-86019bf83fb9%2Fb09d2bdb-310c-46fc-a240-ef84b242634b%2Fz0bc9in_processed.png&w=3840&q=75)
Transcribed Image Text:**Determining Properties of a Given Relation**
**Problem Statement:**
Determine whether the relation defined below is reflexive, symmetric, and/or transitive. Justify your answer.
**Given Relation:**
\[ R \text{ is the relation on } \mathbb{Z} \text{ where } xRy \text{ if } x + y \text{ is even}.\]
*Explanation:*
- **Reflexive**: A relation \(R\) on a set \(A\) is reflexive if every element is related to itself. In other words, for every \(x \in A\), \(xRx\) holds true.
- **Symmetric**: A relation \(R\) on a set \(A\) is symmetric if for every \(x, y \in A\), whenever \(xRy\) holds, \(yRx\) also holds.
- **Transitive**: A relation \(R\) on a set \(A\) is transitive if for every \(x, y, z \in A\), whenever \(xRy\) and \(yRz\) hold, \(xRz\) also holds.
**Justification Requirements:**
- For reflexivity: Verify if \( xRx \) based on the given condition \( x + x = 2x \), and determine if this is always even.
- For symmetry: Check if \( xRy \) implies \( yRx \) based on the given condition \( x + y \).
- For transitivity: Determine if \( xRy \) and \( yRz \) imply \( xRz \) based on the sum properties \( x + y \) and \( y + z \).
This problem involves understanding the mathematical definitions of reflexive, symmetric, and transitive relations, and applying them to the given condition related to even sums.
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