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. theset of nonzero complex numbers. Define a binary relation on this setas follows. Let r₁ cis(0₁) and r2(cis 02) be two elements of C* expressed inpolar form, where 0 ≤ 0 < 2. Thenr₁ 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 threeon a single sketch). Give geometrical descriptions (using words) of eachof these sets (i.e. what can you say about the shape, size, and locationof these three sets?)(c) Give a geometrical description of the equivalence classes of~, in thefollowing form: "The equivalence classes of~, are all _________ centeredat'(d) Based on your description in part (c), show that the equivalence classesof~, form a partition of C*.

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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*.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter5: Rings, Integral Domains, And Fields

Section5.4: Ordered Integral Domains

Problem 1E: Complete the proof of Theorem 5.30 by providing the following statements, where and are arbitrary...

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