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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 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
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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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.)
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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