12. A real number x is called algebraic if there is a positive integer n and integers aŋ, a1, 7 0. Let A be the set of all algebraic numbers. ..., an such that a, + a¡ x + a, x² +· · . +aj x + a2 x +· • · +a, x" = 0 and an (a) Prove that QC A. (b) Prove that /2 e A. (c) Prove that A is countable. Note: You may use the fact that if n is a positive integer, ɑŋ, ɑ1, ..., an are integers, and a, + 0, then {x ER | do + a1 x + a2 x2 +. +a, x" = 0} is finite. ..

12. A real number x is called algebraic if there is a positive integer n and integers aŋ, a1, 7 0. Let A be the set of all algebraic numbers. ..., an such that a, + a¡ x + a, x² +· · . +aj x + a2 x +· • · +a, x" = 0 and an (a) Prove that QC A. (b) Prove that /2 e A. (c) Prove that A is countable. Note: You may use the fact that if n is a positive integer, ɑŋ, ɑ1, ..., an are integers, and a, + 0, then {x ER | do + a1 x + a2 x2 +. +a, x" = 0} is finite. ..

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Description: 12. A real number x is called algebraic if there is a positive integer n andintegers aŋ, a1,7 0. Let A be the set of all algebraic numbers...., an such that a, + a¡ x + a, x² +· · .+aj x + a2 x +· • · +a, x"= 0 and an(a) Prove that QC A.(b) Prove th

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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Statements and Logic

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Question

12. A real number x is called algebraic if there is a positive integer n and
integers aŋ, a1,
7 0. Let A be the set of all algebraic numbers.
..., an such that a, + a¡ x + a, x² +· · .
+aj x + a2 x +· • · +a, x"
= 0 and an
(a) Prove that QC A.
(b) Prove that /2 e A.
(c) Prove that A is countable. Note: You may use the fact that if n is a
positive integer, ɑŋ, ɑ1,
..., an are integers, and a, + 0, then {x ER |
do + a1 x + a2
x2 +.
+a, x" = 0} is finite.
..

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