1. Define a relation on R given by x ~ y if x = y or xy = 1. Prove that equivalence relation. Find the equivalence class of 0, 3 and -2/3. is an
1. Define a relation on R given by x ~ y if x = y or xy = 1. Prove that equivalence relation. Find the equivalence class of 0, 3 and -2/3. is an
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![### Date: Friday, December 1, 2021
1. **Problem 1: Equivalence Relation on Real Numbers**
- Define a relation \( \sim \) on \( \mathbb{R} \) given by \( x \sim y \) if \( x = y \) or \( xy = 1 \). Prove that \( \sim \) is an equivalence relation. Find the equivalence class of 0, 3, and \(-2/3\).
2. **Problem 2: Equivalence Relation on Pairs of Natural Numbers**
- Define a relation \( \sim \) on \( \mathbb{N} \times \mathbb{N} \) by
\[
(n, m) \sim (p, q) \quad \text{if} \quad n + q = p + m.
\]
- Is \( \sim \) an equivalence relation? Why?
3. **Problem 3: Equivalence Relation on Finite Subsets of Natural Numbers**
- Let \( X \) denote the set of all finite subsets of \( \mathbb{N} \), including \(\emptyset\). Define a relation \( \sim \) on elements \( S_1, S_2 \) of \( X \) by \( S_1 \sim S_2 \) if \( S_1 \) and \( S_2 \) have the same cardinality. Show that \( \sim \) is an equivalence relation on \( X \).
4. **Problem 4: Equivalence Relation on Integers**
- Define a relation on \( \mathbb{Z} \) given by \( x \sim y \) if \( x - y \in E \).
- (a) Show that \( \sim \) is an equivalence relation.
- (b) Find the equivalence classes \([0]\sim\) and \([1]\sim\) of 0, 1 \(\in \mathbb{Z}\).
### Explanation Notes
- **Equivalence Relation:** A binary relation \( \sim \) on a set \( S \) is an equivalence relation if it is reflexive, symmetric, and transitive.
- **Equivalence Class:** For an equivalence relation \( \sim \) on a set \( S \), the](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd597ffd2-5c4b-4c2e-8332-77ce1607dac1%2F8c209e04-083a-4cb7-998f-86521f663549%2Fq9d32oa_processed.png&w=3840&q=75)
Transcribed Image Text:### Date: Friday, December 1, 2021
1. **Problem 1: Equivalence Relation on Real Numbers**
- Define a relation \( \sim \) on \( \mathbb{R} \) given by \( x \sim y \) if \( x = y \) or \( xy = 1 \). Prove that \( \sim \) is an equivalence relation. Find the equivalence class of 0, 3, and \(-2/3\).
2. **Problem 2: Equivalence Relation on Pairs of Natural Numbers**
- Define a relation \( \sim \) on \( \mathbb{N} \times \mathbb{N} \) by
\[
(n, m) \sim (p, q) \quad \text{if} \quad n + q = p + m.
\]
- Is \( \sim \) an equivalence relation? Why?
3. **Problem 3: Equivalence Relation on Finite Subsets of Natural Numbers**
- Let \( X \) denote the set of all finite subsets of \( \mathbb{N} \), including \(\emptyset\). Define a relation \( \sim \) on elements \( S_1, S_2 \) of \( X \) by \( S_1 \sim S_2 \) if \( S_1 \) and \( S_2 \) have the same cardinality. Show that \( \sim \) is an equivalence relation on \( X \).
4. **Problem 4: Equivalence Relation on Integers**
- Define a relation on \( \mathbb{Z} \) given by \( x \sim y \) if \( x - y \in E \).
- (a) Show that \( \sim \) is an equivalence relation.
- (b) Find the equivalence classes \([0]\sim\) and \([1]\sim\) of 0, 1 \(\in \mathbb{Z}\).
### Explanation Notes
- **Equivalence Relation:** A binary relation \( \sim \) on a set \( S \) is an equivalence relation if it is reflexive, symmetric, and transitive.
- **Equivalence Class:** For an equivalence relation \( \sim \) on a set \( S \), the
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 2 steps with 2 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
