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2. Define a relation R on Z as follows: a Rb if and only if 5 | (a²-b²). a) Prove that R is an equivalence relation (that is, prove each of the three properties).

2. Define a relation R on Z as follows: a Rb if and only if 5 | (a²-b²). a) Prove that R is an equivalence relation (that is, prove each of the three properties).

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:** 2. Define a relation \( R \) on \( \mathbb{Z} \) as follows: \( a \, R \, b \) if and only if \( 5 \mid (a^2 - b^2) \). a) Prove that \( R \) is an equivalence relation (that is, prove each of the three properties). **Explanation:** To show that \( R \) is an equivalence relation, we need to prove the following properties: 1. **Reflexivity:** For any integer \( a \), it must hold that \( a \, R \, a \). 2. **Symmetry:** For any integers \( a \) and \( b \), if \( a \, R \, b \), then \( b \, R \, a \). 3. **Transitivity:** For any integers \( a \), \( b \), and \( c \), if \( a \, R \, b \) and \( b \, R \, c \), then \( a \, R \, c \).
Transcribed Image Text:**Problem Statement:** 2. Define a relation \( R \) on \( \mathbb{Z} \) as follows: \( a \, R \, b \) if and only if \( 5 \mid (a^2 - b^2) \). a) Prove that \( R \) is an equivalence relation (that is, prove each of the three properties). **Explanation:** To show that \( R \) is an equivalence relation, we need to prove the following properties: 1. **Reflexivity:** For any integer \( a \), it must hold that \( a \, R \, a \). 2. **Symmetry:** For any integers \( a \) and \( b \), if \( a \, R \, b \), then \( b \, R \, a \). 3. **Transitivity:** For any integers \( a \), \( b \), and \( c \), if \( a \, R \, b \) and \( b \, R \, c \), then \( a \, R \, c \).
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