Exercise III.2. Graph the following sets S in R² and determine which are convex: (i) B = : { ( ₁ ) : 2²³² + y² 2² <1} 1){(+) = [IS]A[M<1]} •{(*) : : x² + y² + y² ≤ 1} (iii) (iv) BUC where CC bdry (B) ~{(*) : [le] ≤1] ^ [lu|=1]} (vi) {(7) : [x² + y² <1] ^ [y ≥ 0]} (v) Bo U (ii) B
Exercise III.2. Graph the following sets S in R² and determine which are convex: (i) B = : { ( ₁ ) : 2²³² + y² 2² <1} 1){(+) = [IS]A[M<1]} •{(*) : : x² + y² + y² ≤ 1} (iii) (iv) BUC where CC bdry (B) ~{(*) : [le] ≤1] ^ [lu|=1]} (vi) {(7) : [x² + y² <1] ^ [y ≥ 0]} (v) Bo U (ii) B
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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Transcribed Image Text:Exercise III.2. Graph the following sets \( S \) in \( \mathbb{R}^2 \) and determine which are convex:
(i) \(\mathbf{B} = \left\{ \begin{pmatrix} x \\ y \end{pmatrix} : x^2 + y^2 < 1 \right\}\)
(ii) \(\overline{\mathbf{B}} := \left\{ \begin{pmatrix} x \\ y \end{pmatrix} : x^2 + y^2 \leq 1 \right\}\)
(iii) \(\left\{ \begin{pmatrix} x \\ y \end{pmatrix} : |x| \leq 1 \land |y| < 1 \right\}\)
(iv) \(\mathbf{B} \cup \mathbf{C} \text{ where } \mathbf{C} \subseteq \text{bdry}(\mathbf{B})\)
(v) \(\mathbf{B}_\infty \cup \left\{ \begin{pmatrix} x \\ y \end{pmatrix} : |x| \leq 1 \land |y| = 1 \right\}\)
(vi) \(\left\{ \begin{pmatrix} x \\ y \end{pmatrix} : x^2 + y^2 < 1 \land y \geq 0 \right\}\)
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