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Sketch Complex Regions

  1. Jan 26, 2014 #1
    1. The problem statement, all variables and given/known data
    Sketch the following regions and state the interior and the closure:
    a) |z-2+i|≤1
    b) Im(z)>1


    2. Relevant equations
    z=x+iy


    3. 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?
     
  2. jcsd
  3. Jan 26, 2014 #2

    SammyS

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    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 .
     
  4. Jan 26, 2014 #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?
     
  5. Jan 26, 2014 #4

    SammyS

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    You are confusing a closed set with the closure of a set.
     
  6. Jan 26, 2014 #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?
     
  7. Jan 26, 2014 #6

    SammyS

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    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.
     
  8. Jan 26, 2014 #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.
     
  9. Jan 26, 2014 #8

    SammyS

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    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\ .##
     
  10. Jan 26, 2014 #9
    that really helps! Thanks a lot!
     
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