6. Find all elements of the equivalence class of 12 under congruence modulo 5.
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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![Certainly! Here is the transcription of the given image for educational purposes:
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**6.** Find all elements of the equivalence class of 12 under congruence modulo 5.
**7.** Let \((a, b), (c, d) \in \mathbb{R}^2\). Define a relation \(\sim\) on \(\mathbb{R}^2\) by
\[
(a, b) \sim (c, d) \text{ if } 2a - b = 2c - d.
\]
(a) Show that \(\sim\) is an equivalence relation.
(b) Find the equivalence class of \((0, 1)\).
---
The image contains mathematical exercises dealing with equivalence relations. The first problem requires finding the equivalence class of a number under modular arithmetic, while the second problem involves proving and finding equivalence classes under a defined relation in \(\mathbb{R}^2\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd597ffd2-5c4b-4c2e-8332-77ce1607dac1%2F963c71a7-1dc5-41cb-990c-eb70a54fb6b1%2Fjyfqwwt_processed.png&w=3840&q=75)
Transcribed Image Text:Certainly! Here is the transcription of the given image for educational purposes:
---
**6.** Find all elements of the equivalence class of 12 under congruence modulo 5.
**7.** Let \((a, b), (c, d) \in \mathbb{R}^2\). Define a relation \(\sim\) on \(\mathbb{R}^2\) by
\[
(a, b) \sim (c, d) \text{ if } 2a - b = 2c - d.
\]
(a) Show that \(\sim\) is an equivalence relation.
(b) Find the equivalence class of \((0, 1)\).
---
The image contains mathematical exercises dealing with equivalence relations. The first problem requires finding the equivalence class of a number under modular arithmetic, while the second problem involves proving and finding equivalence classes under a defined relation in \(\mathbb{R}^2\).
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