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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 {r ER" | , Si, Vi = 1, ..., n}. (b) S₂= {xR" x² > i, Vi= 1,...,n}. (c) S3 = {r € R² | 1₁ + x₂ ≤ 2₁ x ² + x ² ≤ 4}. (d) S₁ = {x € Z² | x² + x² ≤ 4}.

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 {r ER" | , Si, Vi = 1, ..., n}. (b) S₂= {xR" x² > i, Vi= 1,...,n}. (c) S3 = {r € R² | 1₁ + x₂ ≤ 2₁ x ² + x ² ≤ 4}. (d) S₁ = {x € Z² | x² + x² ≤ 4}.

Advanced Engineering Mathematics
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) \( S_1 = \{ x \in \mathbb{R}^n \mid x_i \leq i, \forall i = 1, \ldots, n \} \). (b) \( S_2 = \{ x \in \mathbb{R}^n \mid x_i^2 \geq i, \forall i = 1, \ldots, n \} \). (c) \( S_3 = \{ x \in \mathbb{R}^2 \mid x_1 + x_2 \leq 2, x_1^2 + x_2^2 \leq 4 \} \). (d) \( S_4 = \{ x \in \mathbb{Z}^2 \mid x_1^2 + x_2^2 \leq 4 \} \). (e) \( S_5 = \{ x \in \mathbb{R}^2 \mid x_2 \geq e^{x_1} \} \). (f) \( S_6 = \{ x \in \mathbb{R}^2 \mid x_2 \leq e^{x_1} \} \).
Transcribed Image Text: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) \( S_1 = \{ x \in \mathbb{R}^n \mid x_i \leq i, \forall i = 1, \ldots, n \} \). (b) \( S_2 = \{ x \in \mathbb{R}^n \mid x_i^2 \geq i, \forall i = 1, \ldots, n \} \). (c) \( S_3 = \{ x \in \mathbb{R}^2 \mid x_1 + x_2 \leq 2, x_1^2 + x_2^2 \leq 4 \} \). (d) \( S_4 = \{ x \in \mathbb{Z}^2 \mid x_1^2 + x_2^2 \leq 4 \} \). (e) \( S_5 = \{ x \in \mathbb{R}^2 \mid x_2 \geq e^{x_1} \} \). (f) \( S_6 = \{ x \in \mathbb{R}^2 \mid x_2 \leq e^{x_1} \} \).
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