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4. Let R be a relation on the set of all integers. xRy iff x - y = 3m for some integer m. Show that R is an equivalence relation.

4. Let R be a relation on the set of all integers. xRy iff x - y = 3m for some integer m. Show that R is an equivalence relation.

Advanced Engineering Mathematics
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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**Problem Statement:** 4. Let \( R \) be a relation on the set of all integers. \( xRy \) if and only if \( x - y = 3m \) for some integer \( m \). Show that \( R \) is an equivalence relation. --- **Explanation:** To show that \( R \) is an equivalence relation, we need to verify the following properties: 1. **Reflexivity:** For any integer \( x \), \( xRx \) should be true. - Here, \( x - x = 0 \), and 0 can be expressed as \( 3 \times 0 \), which is of the form \( 3m \). Therefore, \( R \) is reflexive. 2. **Symmetry:** For any integers \( x \) and \( y \), if \( xRy \), then \( yRx \). - If \( xRy \), then \( x - y = 3m \), which implies \( y - x = -3m \). Let \( m' = -m \), then \( y - x = 3m' \), showing that \( yRx \). Hence, \( R \) is symmetric. 3. **Transitivity:** For any integers \( x \), \( y \), and \( z \), if \( xRy \) and \( yRz \), then \( xRz \). - If \( xRy \), then \( x - y = 3m_1 \) and if \( yRz \), then \( y - z = 3m_2 \). Adding these, \( x - z = (x - y) + (y - z) = 3m_1 + 3m_2 = 3(m_1 + m_2) \), which is of the form \( 3m \), making \( xRz \). Hence, \( R \) is transitive. Thus, the relation \( R \) is an equivalence relation.
Transcribed Image Text:**Problem Statement:** 4. Let \( R \) be a relation on the set of all integers. \( xRy \) if and only if \( x - y = 3m \) for some integer \( m \). Show that \( R \) is an equivalence relation. --- **Explanation:** To show that \( R \) is an equivalence relation, we need to verify the following properties: 1. **Reflexivity:** For any integer \( x \), \( xRx \) should be true. - Here, \( x - x = 0 \), and 0 can be expressed as \( 3 \times 0 \), which is of the form \( 3m \). Therefore, \( R \) is reflexive. 2. **Symmetry:** For any integers \( x \) and \( y \), if \( xRy \), then \( yRx \). - If \( xRy \), then \( x - y = 3m \), which implies \( y - x = -3m \). Let \( m' = -m \), then \( y - x = 3m' \), showing that \( yRx \). Hence, \( R \) is symmetric. 3. **Transitivity:** For any integers \( x \), \( y \), and \( z \), if \( xRy \) and \( yRz \), then \( xRz \). - If \( xRy \), then \( x - y = 3m_1 \) and if \( yRz \), then \( y - z = 3m_2 \). Adding these, \( x - z = (x - y) + (y - z) = 3m_1 + 3m_2 = 3(m_1 + m_2) \), which is of the form \( 3m \), making \( xRz \). Hence, \( R \) is transitive. Thus, the relation \( R \) is an equivalence relation.
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