Exercise 17.4.12. Consider the set C* defined by C = C\0, i.e. the set of nonzero complex numbers. Define a binary relation on this set as follows. Let r₁ cis(0₁) and r2(cis 02) be two elements of C* expressed in polar form, where 0 ≤ 0 < 27. Then r₁ cis(01) r2(cis 0₂) T1 = 72. (a) Prove that, thus defined is an equivalence relation. (b) Sketch [1], [1 + i], and [ cis(7/3)] in the complex plane (show all three on a single sketch). Give geometrical descriptions (using words) of each of these sets (i.e. what can you say about the shape, size, and location of these three sets?) (c) Give a geometrical description of the equivalence classes of, in the following form: "The equivalence classes of are all at ........". centered (d) Based on your description in part (c), show that the equivalence classes of, form a partition of C*.
Exercise 17.4.12. Consider the set C* defined by C = C\0, i.e. the set of nonzero complex numbers. Define a binary relation on this set as follows. Let r₁ cis(0₁) and r2(cis 02) be two elements of C* expressed in polar form, where 0 ≤ 0 < 27. Then r₁ cis(01) r2(cis 0₂) T1 = 72. (a) Prove that, thus defined is an equivalence relation. (b) Sketch [1], [1 + i], and [ cis(7/3)] in the complex plane (show all three on a single sketch). Give geometrical descriptions (using words) of each of these sets (i.e. what can you say about the shape, size, and location of these three sets?) (c) Give a geometrical description of the equivalence classes of, in the following form: "The equivalence classes of are all at ........". centered (d) Based on your description in part (c), show that the equivalence classes of, form a partition of C*.
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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Please do Exercise 17.4.12 part A,B,C, and D and please show step by step and explain
![Exercise 17.4.12. Consider the set C* defined by C* := C\0, i.e. the
set of nonzero complex numbers. Define a binary relation on this set
as follows. Let r₁ cis(0₁) and r2(cis 02) be two elements of C* expressed in
polar form, where 0 ≤ 0 < 2. Then
r₁ cis(01) r2(cis 02) T1 = 12.
(a) Prove that, thus defined is an equivalence relation.
(b) Sketch [1], [1 + i], and [ cis(/3)] in the complex plane (show all three
on a single sketch). Give geometrical descriptions (using words) of each
of these sets (i.e. what can you say about the shape, size, and location
of these three sets?)
(c) Give a geometrical description of the equivalence classes of~, in the
following form: "The equivalence classes of~, are all _________ centered
at'
(d) Based on your description in part (c), show that the equivalence classes
of~, form a partition of C*.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F892e817a-9b32-4eeb-b8fc-5dd7ffde6479%2Fa35b758c-bb20-461f-836e-7a1a71c678e5%2F0i5kh4r_processed.png&w=3840&q=75)
Transcribed Image Text:Exercise 17.4.12. Consider the set C* defined by C* := C\0, i.e. the
set of nonzero complex numbers. Define a binary relation on this set
as follows. Let r₁ cis(0₁) and r2(cis 02) be two elements of C* expressed in
polar form, where 0 ≤ 0 < 2. Then
r₁ cis(01) r2(cis 02) T1 = 12.
(a) Prove that, thus defined is an equivalence relation.
(b) Sketch [1], [1 + i], and [ cis(/3)] in the complex plane (show all three
on a single sketch). Give geometrical descriptions (using words) of each
of these sets (i.e. what can you say about the shape, size, and location
of these three sets?)
(c) Give a geometrical description of the equivalence classes of~, in the
following form: "The equivalence classes of~, are all _________ centered
at'
(d) Based on your description in part (c), show that the equivalence classes
of~, form a partition of C*.
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