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Magazine Chapter 1: Problem 13
Sketch the set \(S\) of points in the complex plane satisfying the given inequality. Determine whether the set is (a) open, (b) closed, (c) a domain, (d) bounded, or (e) connected. $$ \operatorname{Re}(z)<-1 $$
Short Answer
Expert verified
Set \( S \) is open, not closed, a domain, unbounded, and connected.
Step by step solution
01
Understanding the Inequality
The inequality \( \operatorname{Re}(z) < -1 \) indicates that we are looking for all complex numbers \( z = x + yi \) whose real part \( x \) is less than \(-1\). This means that on the complex plane, we consider the region left to the vertical line \( x = -1 \).
02
Sketch the Set
On the complex plane, draw the vertical line represented by \( x = -1 \). Since the inequality is \( \operatorname{Re}(z) < -1 \), shade the area to the left of this line to denote the set \( S \). The line itself is not included in the set, as indicated by the strict inequality sign "<".
03
Determine Openness
The set \( S \) is open because it includes all points to the left of \( x = -1 \) but not the line itself (the boundary is not included). An open set does not include its boundary.
04
Determine Closedness
Since \( S \) is open and does not contain its boundary \( x = -1 \), it is not a closed set. A closed set includes its boundary.
05
Determine if S is a Domain
A domain in the complex plane is an open and connected set. Since \( S \) is open, we need to check if it is connected. \( S \) is connected because you can reach any point from any other point in \( S \) without crossing the boundary. Thus, \( S \) is a domain.
06
Determine if S is Bounded
A set is bounded if it can be contained within a finite-sized circle. \( S \) extends infinitely to the left and up/down. Therefore, \( S \) is not bounded.
07
Determine if S is Connected
To check if \( S \) is connected, note that there are no separations or holes in the shaded region. Any path within \( S \) does not need to cross any unshaded area. Therefore, \( S \) is connected.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Complex Plane
The complex plane is a geometric representation of complex numbers. Each complex number is expressed as \( z = x + yi \), where \( x \) is the real part and \( y \) is the imaginary part of the number. These are plotted on a two-dimensional plane:
Steps like determining the position of points or their relationships, through inequalities, can easily be analyzed in this space. In our exercise, the focus is on the inequality \( \operatorname{Re}(z) < -1 \), which places complex numbers to the left of the line \( x = -1 \) on the complex plane.
- The horizontal axis represents the real part, labeled as \( x \).
- The vertical axis stands for the imaginary part, labeled as \( y \).
Steps like determining the position of points or their relationships, through inequalities, can easily be analyzed in this space. In our exercise, the focus is on the inequality \( \operatorname{Re}(z) < -1 \), which places complex numbers to the left of the line \( x = -1 \) on the complex plane.
Open Set
An open set in the complex plane is a collection of points that does not include its boundary. This means if you pick any point from the set, there will always be a tiny neighborhood around it entirely contained within the set.
For instance, in our exercise, the set \( S \), defined by the inequality \( \operatorname{Re}(z) < -1 \), is open. It consists of all points where the real part of the complex number is less than \(-1\).
For instance, in our exercise, the set \( S \), defined by the inequality \( \operatorname{Re}(z) < -1 \), is open. It consists of all points where the real part of the complex number is less than \(-1\).
- Since the line \( x = -1 \) is not included, the boundary is not part of the set.
- This is a crucial characteristic of open sets - they do not encapsulate their boundaries.
Domain in Complex Analysis
A domain in complex analysis is a crucial concept involving sets that are both open and connected. The significance of domains lies in their role as the setting for many functions within complex analysis.
In our example:
In our example:
- The set \( S \) is open since it does not contain its boundary.
- It is also connected as any two points within \( S \) can be reached continuously without leaving the set.
Connected Set
A connected set in the complex plane is one where any two points can be connected by a path that lies entirely within the set. When checking for connectedness:
The shaded region extends indefinitely leftward and vertically across any value of \( y \), meaning there is always a continuous way to travel from one point within \( S \) to another.
Connected sets are fundamental in understanding the topology of the complex plane, ensuring no unintended separations in continuous functions or transformations. This connected nature is why \( S \) qualifies as a domain.
- There should be no distinct "holes" or separate pieces in the set.
- Paths between any two points should not require leaving the set.
The shaded region extends indefinitely leftward and vertically across any value of \( y \), meaning there is always a continuous way to travel from one point within \( S \) to another.
Connected sets are fundamental in understanding the topology of the complex plane, ensuring no unintended separations in continuous functions or transformations. This connected nature is why \( S \) qualifies as a domain.
