In each of the cases below, determine whether the set U; is open or not, and brieflyjustify your answers:(i)U1 = {z € C; |z| <1 and |z– 1|<1 and |z- i| < 1}(ii)U2 = {z € C; Re z > 0}(iii)U3 = {z E C; Im z > 0}n{z € C; |z + 10i| < 2}

(i) The given set is U1=z∈ℂ; z<1 and z-1<1 and z-i<1. Define:…

Answered: In each of the cases below, determine…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

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In each of the cases below, determine whether the set U; is open or not, and briefly
justify your answers:
(i)
U1 = {z € C; |z| <1 and |z– 1|<1 and |z- i| < 1}
(ii)
U2 = {z € C; Re z > 0}
(iii)
U3 = {z E C; Im z > 0}n{z € C; |z + 10i| < 2}

Expert Solution

Step 1

(i)

The given set is $U_{1} = \{z \in ℂ; |z| < 1 and |z - 1| < 1 and |z - i| < 1\}$ . Define:

$\begin{array}{rcl} O_{1} & = & \{z \in ℂ : |z| < 1\} \\ O_{2} & = & \{z \in ℂ : |z - 1| < 1\} \\ O_{3} & = & \{z \in ℂ : |z - i| < 1\} \end{array}$

Clearly $U_{1} = O_{1} \cap O_{2} \cap O_{3}$ .

The set $O_{1}$ is an open disc with radius $1$ unit, centered at $z = 0$ and hence open set.

The set $O_{2}$ is an open disc with radius $1$ unit, centered at $z = 1$ and hence open set.

The set $O_{3}$ is an open disc with radius $1$ unit, centered at $z = i$ and hence open set.

Since finite intersection of open set is open in $ℂ$ . Therefore $U_{1} = \{z \in ℂ; |z| < 1 and |z - 1| < 1 and |z - i| < 1\}$ , is an open set.

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In each of the cases below, determine whether the set U; is open or not, and briefly justify your answers: (i) U1 = {z € C; |z| <1 and |z– 1|<1 and |z- i| < 1} (ii) U2 = {z € C; Re z > 0} (iii) U3 = {z E C; Im z > 0}n{z € C; |z + 10i| < 2}