The relation R defined on Z by aRb if 3 | (2a + 7b) is an equivalence relation, I have the equivalence classes but I have no idea how they get them, please explain step by step in detail, I have seen many examples but I really dont get it
The relation R defined on Z by aRb if 3 | (2a + 7b) is an equivalence relation, I have the equivalence classes but I have no idea how they get them, please explain step by step in detail, I have seen many examples but I really dont get it
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
The relation R defined on Z by aRb if 3 | (2a + 7b) is an equivalence relation, I have the equivalence classes but I have no idea how they get them, please explain step by step in detail, I have seen many examples but I really dont get it
![Let's see what the equivalence classes look like for the equivalence relation defined
We begin with the integer 0. Here,
.
In addition,
[0] = {x € Z: x R 0} = {x € Z: 3|(2x+7.0)}
= {x € Z : 3 | 2x} = {x € Z : 3 | x} = {0, ±3, ±6, ...}.
For the integer 1,
[1] = {x € Z: x R 1} = {x € Z : 3 | (2x+7-1)}
= {x € Z: 3 | (2x + 7)} = {1, −2, 4, −5, 7, 8, 10, ...}.
[2] = {x € Z: x R 2} = {x € Z : 3 | (2x+7-2)}
= {x € Z: 3 | (2x + 14)} = {2, —1, 5, —4, 8, —7, 11, ...}.
It turns out that these are the only distinct equivalence classes.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F42044228-cff6-4391-9596-005bad4f5c7f%2F6d94c5f6-332c-4742-9b7d-9f20f922ba2d%2Fmvzqu4q_processed.png&w=3840&q=75)
Transcribed Image Text:Let's see what the equivalence classes look like for the equivalence relation defined
We begin with the integer 0. Here,
.
In addition,
[0] = {x € Z: x R 0} = {x € Z: 3|(2x+7.0)}
= {x € Z : 3 | 2x} = {x € Z : 3 | x} = {0, ±3, ±6, ...}.
For the integer 1,
[1] = {x € Z: x R 1} = {x € Z : 3 | (2x+7-1)}
= {x € Z: 3 | (2x + 7)} = {1, −2, 4, −5, 7, 8, 10, ...}.
[2] = {x € Z: x R 2} = {x € Z : 3 | (2x+7-2)}
= {x € Z: 3 | (2x + 14)} = {2, —1, 5, —4, 8, —7, 11, ...}.
It turns out that these are the only distinct equivalence classes.
Expert Solution
![](/static/compass_v2/shared-icons/check-mark.png)
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 3 steps with 36 images
![Blurred answer](/static/compass_v2/solution-images/blurred-answer.jpg)
Recommended textbooks for you
![Advanced Engineering Mathematics](https://www.bartleby.com/isbn_cover_images/9780470458365/9780470458365_smallCoverImage.gif)
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
![Numerical Methods for Engineers](https://www.bartleby.com/isbn_cover_images/9780073397924/9780073397924_smallCoverImage.gif)
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…](https://www.bartleby.com/isbn_cover_images/9781118141809/9781118141809_smallCoverImage.gif)
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
![Advanced Engineering Mathematics](https://www.bartleby.com/isbn_cover_images/9780470458365/9780470458365_smallCoverImage.gif)
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
![Numerical Methods for Engineers](https://www.bartleby.com/isbn_cover_images/9780073397924/9780073397924_smallCoverImage.gif)
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…](https://www.bartleby.com/isbn_cover_images/9781118141809/9781118141809_smallCoverImage.gif)
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
![Mathematics For Machine Technology](https://www.bartleby.com/isbn_cover_images/9781337798310/9781337798310_smallCoverImage.jpg)
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
![Basic Technical Mathematics](https://www.bartleby.com/isbn_cover_images/9780134437705/9780134437705_smallCoverImage.gif)
![Topology](https://www.bartleby.com/isbn_cover_images/9780134689517/9780134689517_smallCoverImage.gif)