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
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
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter1: Fundamental Concepts Of Algebra
Section1.1: Real Numbers
Problem 40E
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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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