A subset of the real numbers SCR is defined by the statement that x E S whenever x is a real root of a monic quadratic polynomial with integer coefficients, or in other words x satisfies an equation of the form x² +₁x + ₁ = 0 for some integers Co, C₁ € Z. (a) Show that 0 € S and 1 € S. (b) Show that ZCS. (c) Show that Z ‡ S, in other words Z is a proper subset of S.
A subset of the real numbers SCR is defined by the statement that x E S whenever x is a real root of a monic quadratic polynomial with integer coefficients, or in other words x satisfies an equation of the form x² +₁x + ₁ = 0 for some integers Co, C₁ € Z. (a) Show that 0 € S and 1 € S. (b) Show that ZCS. (c) Show that Z ‡ S, in other words Z is a proper subset of S.
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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![A subset of the real numbers SCR is defined by the statement that x E S whenever x is
a real root of a monic quadratic polynomial with integer coefficients, or in other words x
satisfies an equation of the form
x+c+co=0
for some integers Co, C₁ E Z.
(a) Show that 0 € S and 1 € S.
(b) Show that ZC S.
(c) Show that Z ‡ S, in other words Z is a proper subset of S.
(d) Let S'CS be the set of real numbers x that satisfy a quadratic equation of the above
form with coefficients bounded by |co| ≤ d₂ +2, |c₁| ≤ d3 +2, where as before d2, d3 are the
second and third digits of your exam number. Find an upper bound for |S|.
(e) Prove that S is a countable subset of R. (Hint: You may make use of any standard
results on countable sets, provided these are clearly stated.)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F743868de-1649-4a54-9043-62cfa546abf8%2Ff621c1ce-93f2-4f65-bcb3-4f78656bd710%2Fiudhykh_processed.png&w=3840&q=75)
Transcribed Image Text:A subset of the real numbers SCR is defined by the statement that x E S whenever x is
a real root of a monic quadratic polynomial with integer coefficients, or in other words x
satisfies an equation of the form
x+c+co=0
for some integers Co, C₁ E Z.
(a) Show that 0 € S and 1 € S.
(b) Show that ZC S.
(c) Show that Z ‡ S, in other words Z is a proper subset of S.
(d) Let S'CS be the set of real numbers x that satisfy a quadratic equation of the above
form with coefficients bounded by |co| ≤ d₂ +2, |c₁| ≤ d3 +2, where as before d2, d3 are the
second and third digits of your exam number. Find an upper bound for |S|.
(e) Prove that S is a countable subset of R. (Hint: You may make use of any standard
results on countable sets, provided these are clearly stated.)
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