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Magazine Chapter 1: Problem 17
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. $$ 2<\operatorname{Re}(z-1)<4 $$
Short Answer
Expert verified
The set is (a) open, (b) not closed, (c) a domain, (d) not bounded, and (e) connected.
Step by step solution
01
Identify the Inequality
The given inequality is \(2 < \operatorname{Re}(z-1) < 4\). Here, \(z\) is a complex number and represents a point in the complex plane. The expression \(\operatorname{Re}(z-1)\) refers to the real part of \(z-1\). Since \(z = x + yi\) where \(x\) is the real part and \(y\) is the imaginary part, \(\operatorname{Re}(z-1) = x-1\). Therefore, the inequality becomes \(2 < x-1 < 4\).
02
Simplify the Inequality
Simplifying \(2 < x-1 < 4\), we add 1 to each part of the inequality to get \(3 < x < 5\). This means that the real part, \(x\), of any point \(z\) in the set \(S\) must be greater than 3 and less than 5. The inequality places no restriction on the imaginary part, \(y\).
03
Sketch the Region
In the complex plane, the inequality \(3 < x < 5\) represents a vertical strip between the vertical lines at \(x = 3\) and \(x = 5\). The region does not include these lines because the inequality is strict. This area extends infinitely in both directions along the y-axis (imaginary axis).
04
Determine if the Set is Open
A set is open if it does not contain its boundary points. Since the lines \(x = 3\) and \(x = 5\) are not included in the set, the set is open.
05
Determine if the Set is Closed
A set is closed if it contains all its boundary points. In this case, the lines \(x = 3\) and \(x = 5\) are not included, hence the set is not closed.
06
Determine if the Set is a Domain
A domain is an open connected set. Since our set is open and any two points within the strip can be joined by a path that stays within the strip, the set is connected. Hence, it is a domain.
07
Determine if the Set is Bounded
A set is bounded if it fits within some finite radius around a point. Since this strip extends infinitely along the y-axis, the set is not bounded.
08
Determine if the Set is Connected
A set is connected if any two points within the set can be connected by a path that lies entirely within the set. Our set is a vertical strip, which is connected because any point can be reached from any other through a vertical or horizontal path.
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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 two-dimensional plane used to symbolize complex numbers. Think of it like a coordinate plane you use in algebra, but for complex numbers.
- The horizontal axis represents the real part of the complex number.
- The vertical axis represents the imaginary part.
Open Sets
An open set in the complex plane, or any mathematical space, is a set that does not include its boundary points. Consider the inequality \(2 < \operatorname{Re}(z-1) < 4\), which simplifies to \(3 < x < 5\). This inequality forms a vertical strip where the real part is between 3 and 5.
- The boundary of this strip would be on the lines \(x = 3\) and \(x = 5\).
- These lines are not included in the set described by the inequality due to the strict inequality signs.
Connected Sets
A connected set is one in which any two points can be joined by a path entirely contained within the set. In the complex plane, the region defined by \(3 < x < 5\) forms a vertical strip.
- This strip is tall, reaching infinitely up and down the imaginary axis.
- Within this region, you can move between any two points by a path consisting solely of movements within the region.
Inequality in Complex Numbers
Inequalities help describe regions in the complex plane. In our case with \(2 < \operatorname{Re}(z-1) < 4\), the focus is on the real part of the complex number.
- The inequality simplifies to \(3 < x < 5\), indicating an interval on the real axis where the real parts of the numbers lie.
- The imaginary part, \(y\), does not affect this inequality, allowing it to vary freely.
- This specifies a region in the plane, bound between two vertical lines, but not including them.
