4. Show that it is impossible to define a total ordering on C. In other words, one cannot find a relation between complex numbers so that: (i) For any two complex numbers z, w, one and only one of the following is true: zyw, wy z or 2 = w. (ii) For all 21, 22, 23 € C the relation 21 22 implies 2₁+ 23 22 +23. (iii) Moreover, for all 21, 22, 23 E C with 230, then 21 22 implies z1 23 ➤ 2223- [Hint: First check if i0 is possible.]
4. Show that it is impossible to define a total ordering on C. In other words, one cannot find a relation between complex numbers so that: (i) For any two complex numbers z, w, one and only one of the following is true: zyw, wy z or 2 = w. (ii) For all 21, 22, 23 € C the relation 21 22 implies 2₁+ 23 22 +23. (iii) Moreover, for all 21, 22, 23 E C with 230, then 21 22 implies z1 23 ➤ 2223- [Hint: First check if i0 is possible.]
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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![4. Show that it is impossible to define a total ordering on C. In other words, one
cannot find a relation between complex numbers so that:
(i) For any two complex numbers z, w, one and only one of the following is true:
zyw, wy z or 2=w.
(ii) For all 21, 22, 23 € C the relation 21 22 implies 2₁+ 23 > 22 +23.
(iii) Moreover, for all 21, 22, 23 E C with 230, then 21 22 implies z1 23 ➤ 2223-
[Hint: First check if i0 is possible.]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F904caeb1-cd31-4dcf-9af8-09f5b2d51865%2F66478740-ea7b-4fe9-8c4c-332cac99f3c0%2F9c7vszg_processed.jpeg&w=3840&q=75)
Transcribed Image Text:4. Show that it is impossible to define a total ordering on C. In other words, one
cannot find a relation between complex numbers so that:
(i) For any two complex numbers z, w, one and only one of the following is true:
zyw, wy z or 2=w.
(ii) For all 21, 22, 23 € C the relation 21 22 implies 2₁+ 23 > 22 +23.
(iii) Moreover, for all 21, 22, 23 E C with 230, then 21 22 implies z1 23 ➤ 2223-
[Hint: First check if i0 is possible.]
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