Exercise 17.5.15. Find x, y € Z12 such that a 0 and y = 0, but x - y = 0.
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
Please do Exercise 17.5.15 and please show step by step and explain
![**Proposition 17.5.12.** For any \( n \in \mathbb{N}^+ \), we have
\[ Z_n = \{ \overline{0}, \overline{1}, \overline{2}, \ldots, \overline{n-1} \} \]
and \( \overline{0}, \overline{1}, \overline{2}, \ldots, \overline{n-1} \) are all distinct.
**Exercise 17.5.13.** Prove Proposition 17.5.12. It is sufficient to show (a) \( \overline{0}, \overline{1}, \overline{2}, \ldots, \overline{n-1} \) are distinct; and (b) for any integer, the equivalence class \( \overline{k} \) is one of \( \overline{0}, \overline{1}, \overline{2}, \ldots, \overline{n-1} \).
♦
**Exercise 17.5.14.** Using the definitions of addition, subtraction, and multiplication given in part (c) of Definition 17.5.11, make tables that show the results of:
(a) addition modulo 4.
(b) subtraction modulo 5.
(c) multiplication modulo 6.
**Exercise 17.5.15.** Find \( x, y \in Z_{12} \) such that \( x \neq \overline{0} \) and \( y \neq \overline{0} \), but \( x \cdot y = \overline{0} \).
♦](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F892e817a-9b32-4eeb-b8fc-5dd7ffde6479%2Fa67f1f51-9e7c-406d-8aba-e2402f014b2e%2Fawb49jg_processed.png&w=3840&q=75)
Transcribed Image Text:**Proposition 17.5.12.** For any \( n \in \mathbb{N}^+ \), we have
\[ Z_n = \{ \overline{0}, \overline{1}, \overline{2}, \ldots, \overline{n-1} \} \]
and \( \overline{0}, \overline{1}, \overline{2}, \ldots, \overline{n-1} \) are all distinct.
**Exercise 17.5.13.** Prove Proposition 17.5.12. It is sufficient to show (a) \( \overline{0}, \overline{1}, \overline{2}, \ldots, \overline{n-1} \) are distinct; and (b) for any integer, the equivalence class \( \overline{k} \) is one of \( \overline{0}, \overline{1}, \overline{2}, \ldots, \overline{n-1} \).
♦
**Exercise 17.5.14.** Using the definitions of addition, subtraction, and multiplication given in part (c) of Definition 17.5.11, make tables that show the results of:
(a) addition modulo 4.
(b) subtraction modulo 5.
(c) multiplication modulo 6.
**Exercise 17.5.15.** Find \( x, y \in Z_{12} \) such that \( x \neq \overline{0} \) and \( y \neq \overline{0} \), but \( x \cdot y = \overline{0} \).
♦
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
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,
