1. For each of the following sets, select all that are convex:(a) S = {(x1, x2) € R² | x² + x² ≤ 4, x1 ≥ 0, x2 ≤ 0}.(b) S = : {(x₁, x₂) = Z²|x² + x² ≤ 4, x₁ ≥ 0, x2 ≤ 0}.(c) S = {x € R¹| Σ1 x² ≤ 4, x₁ ≥ 0, x2 ≤ 0}.(d) S = {x = R¹ |x₁ + x2 = 10(x3 + x4)}, where n ≥ 4.(e) S = {x ER | maxi=1,...,nnhi10 mini 1,..., {i}}.=

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Answered: 1. For each of the following sets,…

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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2. Consider the following sets. Determine which one(s) are convex and which one(s) are not. For any set that is not convex, give two points that show this set is not convex. For any set that is convex, explain why using one of the rules seen in class or using first principles: (a) S₁ = {x € R" | x₁ ≤i, Vi= 1,...,n}. (b) S₂ = {x ER | a? > i, Vi= 1,...,n}. (c) $g= { € R2 \x1+≤2, i+ <4}. (d) S₁ = {x € Z² | x² + x² ≤ 4}. (e) S5 = {x R² | 1₂ ≥ e¹¹}. (f) S6 = {x € R² | x₂ ≤e²²}.
Which of the following sets are well-ordered? Justify your answers. (a) A = {r € Q : r > -5} (b) A = {-4, –3, –2, –1,0, 1, 2, 3, 4} (c) A = {r € Q : -2
1.47. Which of the following sets are equal? A = {x:x² - 4x + 3 = 0}, B = {x:x²-3x + 2 = 0}, C = (x:xEN, x <3}, D= (x:xEN, x is odd, x < 5}, E = {1,2}, F = {1,2, 1}, G=(3, 1), H = {1, 1,3} 1.48. List the elements of the following sets if the universal set is the English alphabet U = {a, b, c,..., y, z}. Furthermore, identify which of the sets are equal. A = {x:x is a vowel}, B = {x:x is a letter in the word "little"}, C = {x:x precedes f in the alphabet}, D = {x: x is a letter in the word "title"}
Show the following set is convex. x10 Determine a-cut sets of the above set for a=0.5, 0.8 and 0.9.
1. Determine the following sets. The symbols An and An mean to take the intersection or union of the sets A₁, A2,... [0,1/n] (a) (b) WL ∞ n=1 ∞ (0, 1/n) n=1 n=1 n=1 Recall that in interval notation [a, b] = {x\a ≤ x ≤ b]. Recall that in interval notation (a, b) = {x\a < x < b]. fab
F and H are sets of real numbers defined as follows. F={y ]y21} H= {y]y>7} Write FUH and FnHusing interval notation. If the set is empty, write Ø. (0,0) (0,0) (0,0) FUH = [0,0) DUO FnH =D 00 -00
(c) Show that the set {r EN | x² % 4 = 1} is equal to the set {r EN | x² % 8 = 1}.
Answer ONLY part C
Which of the following pairs of sets are equivalent or equal?

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1. For each of the following sets, select all that are convex: (a) S = {(1, ₂) € R² | x² + x² ≤ 4, x₁ ≥ 0, x₂ ≤ 0}. (b) S = {(x1, x2) € ZZ²|x² + x² ≤ 4, x₁ ≥ 0, x2 ≤ 0}. (c) S = {xe R¹|1x² ≤ 4, x₁ ≥ 0,02 ≤ 0}.