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QUESTION Sketch the following sets and determine which are
regions.
\begin{description}
\item[(a)]
$|z-2+i|\leq 1.$
\item[(b)]
$|2z+3|>4.$
\item[(c)]
Im$z>3$
\item[(d)]
$|z-4|\geq|z|$
\item[(e)]
$0\leq\textrm{Arg} z\leq \frac{\pi}{4},\ (z\neq0.)$
\end{description}
ANSWER
\begin{description}
\item[(a)]
$|z-2+1|\leq 1$ is a disk center $2-i$, radius 1. As it includes
the boundary, it is not open and so not a region. (in fact it is a
closed set.)
\item[(b)]
$|2z+3|>4\Leftrightarrow |z+\frac{3}{2}|>2$. This defines a region
exterior to a disc centre $-\frac{3}{2}$, radius 2. It does not
include the boundary so it is an open set. Also it is connected
and hence a region.
\item[(c)]
Im$z>3$ is a half-plane not including the line Im$z=3$ so it is
open and connected and hence a region.
\item[(d)]
This is the set of points closer to zero than to 4 (including the
line $x=2$.) Thus it is the half-plane on the side of the line
$x=2$ containing 0. As it includes the line $x=2$ it is not open
and thus not a region.
\item[(e)]
This set contains the line arg$z=\frac{\pi}{4}$ so it is not open
and hence not a region. Actually, its complement contains 0 but no
neighbourhood of 0 so it's complement is not open so the set is
also not closed.
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