Exercise III.2. Graph the following sets S in R² and determine which are conver: (i) B (ii) B:= • { ( ₁ ) : 2²³² + √² ≤ ² {(); ²² +*² <1} [la] ≤1] ^ [<1]} (iii) : (iv) BUC where CC bdry(B) (v) Bo ~{(*): [|~| ≤1] ^ [y] =1]} (vi) {{"): [2² + y² <1] ^ [y ≥0]}
Exercise III.2. Graph the following sets S in R² and determine which are conver: (i) B (ii) B:= • { ( ₁ ) : 2²³² + √² ≤ ² {(); ²² +*² <1} [la] ≤1] ^ [<1]} (iii) : (iv) BUC where CC bdry(B) (v) Bo ~{(*): [|~| ≤1] ^ [y] =1]} (vi) {{"): [2² + y² <1] ^ [y ≥0]}
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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I just need to know which ones are convex, no need to graph. Thank you!
![**Exercise III.2.** Graph the following sets \( S \) in \( \mathbb{R}^2 \) and determine which are convex:
(i) \[ \textbf{B} = \left\{ \begin{pmatrix} x \\ y \end{pmatrix} : x^2 + y^2 < 1 \right\} \]
(ii) \[ \bar{\textbf{B}} := \left\{ \begin{pmatrix} x \\ y \end{pmatrix} : x^2 + y^2 \leq 1 \right\} \]
(iii) \[ \left\{ \begin{pmatrix} x \\ y \end{pmatrix} : |x| \leq 1 \wedge |y| < 1 \right\} \]
(iv) \[ \textbf{B} \cup \textbf{C} \text{ where } \textbf{C} \subseteq \text{bdry}(\textbf{B}) \]
(v) \[ \textbf{B}_\infty \cup \left\{ \begin{pmatrix} x \\ y \end{pmatrix} : |x| \leq 1 \wedge |y| = 1 \right\} \]
(vi) \[ \left\{ \begin{pmatrix} x \\ y \end{pmatrix} : x^2 + y^2 < 1 \wedge y \geq 0 \right\} \]
Note:
- \(\textbf{B}\) and \(\bar{\textbf{B}}\) represent the open and closed unit disks in the plane, respectively.
- \(\textbf{B}_\infty\) denotes the unit ball in the \(L^\infty\) norm, typically a square of side 2 centered at the origin with sides parallel to the axes.
- \(\text{bdry}(\textbf{B})\) refers to the boundary of set \(\textbf{B}\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe87a569b-7721-4104-9b29-fedcf0a2817b%2Fcfb72184-673f-4143-8deb-f38922ebf178%2Flpjzfsn_processed.png&w=3840&q=75)
Transcribed Image Text:**Exercise III.2.** Graph the following sets \( S \) in \( \mathbb{R}^2 \) and determine which are convex:
(i) \[ \textbf{B} = \left\{ \begin{pmatrix} x \\ y \end{pmatrix} : x^2 + y^2 < 1 \right\} \]
(ii) \[ \bar{\textbf{B}} := \left\{ \begin{pmatrix} x \\ y \end{pmatrix} : x^2 + y^2 \leq 1 \right\} \]
(iii) \[ \left\{ \begin{pmatrix} x \\ y \end{pmatrix} : |x| \leq 1 \wedge |y| < 1 \right\} \]
(iv) \[ \textbf{B} \cup \textbf{C} \text{ where } \textbf{C} \subseteq \text{bdry}(\textbf{B}) \]
(v) \[ \textbf{B}_\infty \cup \left\{ \begin{pmatrix} x \\ y \end{pmatrix} : |x| \leq 1 \wedge |y| = 1 \right\} \]
(vi) \[ \left\{ \begin{pmatrix} x \\ y \end{pmatrix} : x^2 + y^2 < 1 \wedge y \geq 0 \right\} \]
Note:
- \(\textbf{B}\) and \(\bar{\textbf{B}}\) represent the open and closed unit disks in the plane, respectively.
- \(\textbf{B}_\infty\) denotes the unit ball in the \(L^\infty\) norm, typically a square of side 2 centered at the origin with sides parallel to the axes.
- \(\text{bdry}(\textbf{B})\) refers to the boundary of set \(\textbf{B}\).
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