5. Find all x € Z satisfying the congruence equation x = 1mod 5. 6. Find all elements of the equivalence class of 12 under congruence modulo 5. 7. Let (a, b), (c,d) € R². Define a relation ~on R² by (a, b)~ (c,d) if 2a − b = 2c - d. (a) Show that ~ is an equivalence relation. (b) Find the equivalence class of (0, 1).
5. Find all x € Z satisfying the congruence equation x = 1mod 5. 6. Find all elements of the equivalence class of 12 under congruence modulo 5. 7. Let (a, b), (c,d) € R². Define a relation ~on R² by (a, b)~ (c,d) if 2a − b = 2c - d. (a) Show that ~ is an equivalence relation. (b) Find the equivalence class of (0, 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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Question
![### Mathematical Concepts and Problems
5. **Congruence Equation Problem**
- **Task:** Find all \( x \in \mathbb{Z} \) satisfying the congruence equation \( x \equiv 1 \pmod{5} \).
6. **Equivalence Class under Modulo Operation**
- **Task:** Find all elements of the equivalence class of 12 under congruence modulo 5.
7. **Equivalence Relation on \(\mathbb{R}^2\)**
- **Definition:** Let \((a, b), (c, d) \in \mathbb{R}^2\). Define a relation \(\sim\) on \(\mathbb{R}^2\) by
\[
(a, b) \sim (c, d) \text{ if } 2a - b = 2c - d.
\]
- **Tasks:**
- (a) Show that \(\sim\) is an equivalence relation.
- (b) Find the equivalence class of \((0, 1)\).
8. **Equivalence Relation on \(\mathbb{R}\)**
- **Definition:** Define a relation ‘\(\sim\)’ on \(\mathbb{R}\) by
\[
x \sim y \iff x - y \in \mathbb{Z}
\]
- **Tasks:**
- (a) Show that \(\sim\) is an equivalence relation.
- (b) Prove that the equivalence class \([0]_\sim\) of \(0 \in \mathbb{R}\) is the set \(\mathbb{Z}\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd597ffd2-5c4b-4c2e-8332-77ce1607dac1%2Fa8591cdd-8717-4e37-becb-380eef4dd170%2Fk2w7bpu_processed.png&w=3840&q=75)
Transcribed Image Text:### Mathematical Concepts and Problems
5. **Congruence Equation Problem**
- **Task:** Find all \( x \in \mathbb{Z} \) satisfying the congruence equation \( x \equiv 1 \pmod{5} \).
6. **Equivalence Class under Modulo Operation**
- **Task:** Find all elements of the equivalence class of 12 under congruence modulo 5.
7. **Equivalence Relation on \(\mathbb{R}^2\)**
- **Definition:** Let \((a, b), (c, d) \in \mathbb{R}^2\). Define a relation \(\sim\) on \(\mathbb{R}^2\) by
\[
(a, b) \sim (c, d) \text{ if } 2a - b = 2c - d.
\]
- **Tasks:**
- (a) Show that \(\sim\) is an equivalence relation.
- (b) Find the equivalence class of \((0, 1)\).
8. **Equivalence Relation on \(\mathbb{R}\)**
- **Definition:** Define a relation ‘\(\sim\)’ on \(\mathbb{R}\) by
\[
x \sim y \iff x - y \in \mathbb{Z}
\]
- **Tasks:**
- (a) Show that \(\sim\) is an equivalence relation.
- (b) Prove that the equivalence class \([0]_\sim\) of \(0 \in \mathbb{R}\) is the set \(\mathbb{Z}\).
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