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Exercise 1.4.10 (Challenging): Let F be the set of all functions f: R→R. Prove |R| < |F| using Cantor's Theorem 0.3.34.*

Exercise 1.4.10 (Challenging): Let F be the set of all functions f: R→R. Prove |R| < |F| using Cantor's Theorem 0.3.34.*

Advanced Engineering Mathematics
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 solve Exercise 1.4.10 with detailed explanations

Exercise 1.4.9 (Challenging): A number x is algebraic if x is a root of a polynomial with integer coefficients, in other words, anx" +an_1x²-1 +...+a₁x+ag=0 where all an € Z. a) Show that there are only countably many algebraic numbers. b) Show that there exist non-algebraic (transcendental) numbers (follow in the footsteps of Cantor, use the uncountability of R). Hint: Feel free to use the fact that a polynomial of degree n has at most n real roots. Exercise 1.4.10 (Challenging): Let F be the set of all functions f: R→ R. Prove |R|< |F| using Cantor's Theorem 0.3.34.*
Transcribed Image Text:Exercise 1.4.9 (Challenging): A number x is algebraic if x is a root of a polynomial with integer coefficients, in other words, anx" +an_1x²-1 +...+a₁x+ag=0 where all an € Z. a) Show that there are only countably many algebraic numbers. b) Show that there exist non-algebraic (transcendental) numbers (follow in the footsteps of Cantor, use the uncountability of R). Hint: Feel free to use the fact that a polynomial of degree n has at most n real roots. Exercise 1.4.10 (Challenging): Let F be the set of all functions f: R→ R. Prove |R|< |F| using Cantor's Theorem 0.3.34.*
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