Which of the given sets are closed regions arg z Solution1ar

Which of the given sets are closed regions? |arg z|

Solution

1)arg(z)arg(z) is the angle between the line joining your point z=a+ibz=a+ib and the origin, in the complex plane. It becomes visually striking when you are introduced to the polar form of complex numbers, but from the situation you describe, you have just started working on complex numbers.This region is (simply) connected, as you can partition it in as many subsets as you want, and it is open, as it extends to infinity (i.e. it is not completely bounded)

2) |z-1+i| =|x+iy-1+i|=|(x-1)+(y+1)i | =sq.root (x-1)^2 +(y+1)^2<= 9 which is a circle with centre (1,-1) this circle is bounded hence the given set is bounded

3) it is not a bounded set because it does not contain all the boundary points . -1 is not in the set

4) also D,E,F are not bounded because they does not contain the boundary points