Determine if the given set is Bounded- Complex Numbers

In summary, the conversation discusses the concept of radius in a mathematical context, specifically in relation to a given sequence that is both bounded and monotonic. The individual struggles to understand why the radius is equal to 2 and mentions that the boundary of the set in question includes the complement of the set. The conversation also brings up different regions and sets that are related to the given sequence. The expert provides clarification on the concept of set membership and how it relates to the radius.
  • #1
chwala
Gold Member
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Homework Statement
See attached.
Relevant Equations
Complex Numbers
My interest is only on part (a). Wah! been going round circles to try understand why the radius = ##2##. I know that the given sequence is both bounded and monotonic. I can state that its bounded above by ##1## and bounded below by ##0##. Now when it comes to the radius=##2##, i can also say that the boundary of set ##S## also consists/ includes the complement of the set and that will gives us;##r^2=\sqrt{(1--1)^2+(0-0)^2}=\sqrt{4}##
##r=2.##

I hope this is the correct reasoning, otherwise i need your insight...i also tried looking at the cauchy criterion,... among other...

1666698269601.png
 
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  • #2
Say s is a member of S other than i , as ##|s| < 1## s is inside unit circle of complex plane on which i is. Both s and i are inside the circle |z|=2.

Not only |z|##\leq##2 but other regions including S inside work, e.g. |z|##\leq##3/2, |z|##\leq##3,
the rectangle region of ## 0 \leq I am \ z \leq 1+\epsilon_1, |Re\ z| \leq \epsilon_2 ## where ##0<\epsilon_1,\epsilon_2##.
2022-10-25 22.56.34.jpg
 
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  • #3
anuttarasammyak said:
Say s is a member of S other than i , as ##|s| < 1## s is inside unit circle of complex plane on which i is. Both s and i are inside the circle |z|=2.

Not only |z|##\leq##2 but other regions including S inside work, e.g. |z|##\leq##3/2, |z|##\leq##3,
the rectangle region of ## 0 \leq I am \ z \leq 1+\epsilon_1, |Re\ z| \leq \epsilon_2 ## where ##0<\epsilon_1,\epsilon_2##.
View attachment 316045
Something am not getting here, I guess I need to check...I thought the set members are bound by ##1## and ##0##. You have ##1.5## which clearly in my understanding is not part of set S. My understanding of radius here is the distance between i.e neighbourhood of set ##S## and ##S^{'}##.
 
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  • #4
chwala said:
You have 1.5 which clearly in my understanding is not part of set S.
You are right. Even as for the answer in the textbook, z=1 belongs to |z|<2, but it does not belong to set S.

Say T is a set, e.g. |z| ##\leq## 2, |z| ##\leq \sqrt{2}##, |z| ##\leq \pi##, |z| ##\leq 3/2## ... any circle larger than unit circle in complex plane,
##S \subset T##
Set T is not part of set S. Set S is part of set T.
 
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Related to Determine if the given set is Bounded- Complex Numbers

1. What does it mean for a set to be bounded in complex numbers?

For a set to be bounded in complex numbers, it means that all the elements in the set have a finite distance from the origin in the complex plane. This can also be thought of as the set being contained within a certain radius or distance from the origin.

2. How can I determine if a given set is bounded in complex numbers?

To determine if a given set is bounded in complex numbers, you can plot the elements of the set on a complex plane and check if they are contained within a certain radius from the origin. Alternatively, you can also calculate the modulus or absolute value of each element and check if they are all less than a certain value.

3. Is the empty set bounded in complex numbers?

Yes, the empty set is considered bounded in complex numbers since it does not contain any elements and therefore all elements (or lack thereof) have a finite distance from the origin.

4. Can a set be partially bounded in complex numbers?

Yes, a set can be partially bounded in complex numbers if some elements are contained within a certain radius from the origin, while others are not. In this case, the set can be considered bounded up to a certain point or value.

5. What is an unbounded set in complex numbers?

An unbounded set in complex numbers is a set where at least one element does not have a finite distance from the origin. This means that the set extends infinitely in at least one direction in the complex plane.

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