A. Give the Dedekind cuts in ℝ ≥0 corresponding to the following. Your definition should not refer to the elements themselves.i. √6 ii. 3√5 iii. √2 + √5B. Show that multiplication of two Dedekind cuts in ℝ ≥0(as on p17 of the notes) is commutative and associative.C. Prove that F2 (as defined on p20 of the notes) is a field.D. Let S and T be sets of real numbers, and define S + T to be {x + y : x ∈ S and y ∈ T}.Show that if S and T are both bounded, then S + T is also bounded.E. Prove that the set of numbers ℚ(√3) = { a + b√3 : a, b ∈ ℚ } is a field. (You may use that ℝ is a field; this makes checking e.g. associativity very easy!)

We have to find Dedekind cuts in ℝ ≥0 for the following: (i) 6. (ii) 53. (iii) 2+5. As per policy,…

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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. Give the Dedekind cuts in ℝ^≥0 corresponding to the following. Your definition should not refer to the elements themselves.
i. √6 ii. ³√5 iii. √2 + √5

B. Show that multiplication of two Dedekind cuts in ℝ^≥0(as on p17 of the notes) is commutative and associative.

C. Prove that F₂ (as defined on p20 of the notes) is a field.

D. Let S and T be sets of real numbers, and define S + T to be
{x + y : x ∈ S and y ∈ T}.
Show that if S and T are both bounded, then S + T is also bounded.

E. Prove that the set of numbers ℚ(√3) = { a + b√3 : a, b ∈ ℚ } is a field. (You may use that ℝ is a field; this makes checking e.g. associativity very easy!)

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