Sketch Complex Regions: |z-2+i|≤1 and Im(z)>1

In summary: The closure of a set ##\ X\ ## can be thought of as the "smallest" closed set which contains ##\ X\ .##that really helps! Thanks a lot!
  • #1
alexcc11
21
0

Homework Statement


Sketch the following regions and state the interior and the closure:
a) |z-2+i|≤1
b) Im(z)>1


Homework Equations


z=x+iy


The Attempt at a Solution


a) z=x+iy so |x+iy-2+i|-> |(x-2)+i(y+1)|≤1
So (x-2)2+(y+1)2≤1

So it would just be a circle on the real plane? And the interior would be the equation with < instead of ≤ right? I'm not sure how to write the closure though.

b)Im(z)= y so it would be a straight line at y=1 on the real plane and there is no interior or closure since it is an open set right?
 
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  • #2
alexcc11 said:

Homework Statement


Sketch the following regions and state the interior and the closure:
a) |z-2+i|≤1
b) Im(z)>1


Homework Equations


z=x+iy


The Attempt at a Solution


a) z=x+iy so |x+iy-2+i|-> |(x-2)+i(y+1)|≤1
So (x-2)2+(y+1)2≤1

So it would just be a circle on the real plane? And the interior would be the equation with < instead of ≤ right? I'm not sure how to write the closure though.

b)Im(z)= y so it would be a straight line at y=1 on the real plane and there is no interior or closure since it is an open set right?
What is the definition of the closure of a set?


For (b):

y = 1 is the solution to Im(z) = 1, not Im(z) > 1 .
 
  • #3
My book says that a set is closed if it contains all boundary points, but Im(z)>1 is open, so there is no closure. Do you know if the other parts are correct, referring to graphing them?
 
  • #4
alexcc11 said:
My book says that a set is closed if it contains all boundary points, but Im(z)>1 is open, so there is no closure. Do you know if the other parts are correct, referring to graphing them?
You are confusing a closed set with the closure of a set.
 
  • #5
How do you show the closure of a set if it's an open set? For part a, would the showing the closure just be writing the equation out since it's a closed set?
 
  • #6
alexcc11 said:
How do you show the closure of a set if it's an open set? For part a, would the showing the closure just be writing the equation out since it's a closed set?

First of all, does your book give the definition of the closure of a set?

Secondly, your answer to (a) was correct, since the closure of a closed set is the set itself.
 
  • #7
The book just says Ω is closed if Ω <=> C'Ω where Ω in C'Ω has a line above it and = {zεC: each neighborhood = D_ε(z) intersects Ω

An example is: E={zεC: .5≤|z|<1}

E_1= {zεC :|z|=1/2}
E_2= {zεC :|z|=1}
E_3= {zεC : .5<|z|<1}

The closure is:E_1 [itex]\bigcup[/itex]E_2 [itex]\bigcup[/itex]E_3

but I don't know who to write that with these problems.
 
  • #8
alexcc11 said:
The book just says Ω is closed if Ω <=> C'Ω where Ω in C'Ω has a line above it and = {zεC: each neighborhood = D_ε(z) intersects Ω

An example is: E={zεC: .5≤|z|<1}

E_1= {zεC :|z|=1/2}
E_2= {zεC :|z|=1}
E_3= {zεC : .5<|z|<1}

The closure is:E_1 [itex]\bigcup[/itex]E_2 [itex]\bigcup[/itex]E_3

but I don't know who to write that with these problems.

Nothing there gives a definition of the closure of a set.

The given example can be helpful.

The set E itself is neither closed nor open.

Notice that ##\ E_3\ ## is open and is the interior of ##\ E\ .##

Furthermore, ##\ E_1\cup E_2\ ## is the boundary of ##\ E\ .##

One can also say that ##\ E\cup E_2\ ## is closed. It's also the closure of ##\ E\ .##

The closure of a set ##\ X\ ## can be thought of as the "smallest" closed set which contains ##\ X\ .##
 
  • #9
that really helps! Thanks a lot!
 

Related to Sketch Complex Regions: |z-2+i|≤1 and Im(z)>1

1. What are complex regions in a sketch?

Complex regions in a sketch refer to areas that have multiple boundaries or curves, making them difficult to define or represent with simple shapes. These regions often have irregular or intricate shapes that cannot be easily drawn or measured.

2. How can I identify complex regions in a sketch?

Complex regions can be identified by looking for areas with multiple intersecting lines, curved edges, or irregular shapes. These regions may also have multiple layers or overlapping shapes, making them difficult to define or measure accurately.

3. Why is it important to accurately sketch complex regions?

Accurately sketching complex regions is important because it ensures that the final measurement or representation of the region is as precise and reliable as possible. Inaccurate sketches can lead to incorrect measurements, which may affect the overall results or conclusions of a scientific study or experiment.

4. What are some techniques for sketching complex regions?

Some techniques for sketching complex regions include breaking the region into smaller, simpler shapes, using reference points or measurements to define boundaries, and using tools such as a compass or protractor to accurately draw curves or angles.

5. Are there any software tools that can assist with sketching complex regions?

Yes, there are various software tools that can assist with sketching complex regions, such as computer-aided design (CAD) software or image editing programs. These tools often have features that allow for precise measurements and drawing of complex shapes, making it easier to accurately represent these regions in a sketch.

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