Problem 27 - Calculus Early Transcendentals: Pearson New International Edition

Question

Sketch the indicated set. Describe the boundary of the set. Finally, state whether the set is open, closed, or neither. \(\\{(x, y): 2 \leq x \leq 4,1 \leq y \leq 5\\}\)

Step-by-Step Solution

Answer

The set is closed and forms a rectangle including its boundary.

Step 1: Understand the Set

The given set \(\{(x, y): 2 \leq x \leq 4, 1 \leq y \leq 5\}\) represents all points \((x, y)\) in a coordinate plane where \(x\) is between 2 and 4 (inclusive), and \(y\) is between 1 and 5 (inclusive). This means each \(x\) and \(y\) will be a point in this range, including the boundaries at \(x=2\), \(x=4\), \(y=1\), and \(y=5\).

Step 2: Sketch the Set on the Coordinate Plane

To sketch the set, draw a rectangle in the Cartesian plane. The bottom-left corner of the rectangle is \((2, 1)\) and the top-right corner is \((4, 5)\). The edges of the rectangle match the boundaries of \(x\) and \(y\), so all points within this area, including those on the edges and corners, are part of the set.

Step 3: Describe the Boundary

The boundary of the set is the perimeter of the rectangle formed. This includes all the points along the lines \(x=2\), \(x=4\), \(y=1\), and \(y=5\) within the given range. These lines are solid because they are included in the set.

Step 4: Determine if the Set is Open, Closed, or Neither

An open set does not include its boundary points, while a closed set does. Since all boundary points of the rectangle \( (2, 1) \) to \( (4, 5) \) are included (because of the \( \leq \) signs), the set is closed. Therefore, the set is closed as it includes its boundary.

Key Concepts

Set Theory

Set theory is a branch of mathematical logic that studies sets, which are collections of objects. In simple terms, a set is a well-defined collection of distinct objects, considered as an object in its own right. Sets are denoted by curly braces. For example, the set \( \{1, 2, 3\} \) contains the numbers 1, 2, and 3.
In coordinate geometry, such sets often refer to collections of points that satisfy specific conditions. The given exercise introduces a set comprising points within a given rectangular region of the Cartesian plane. This set is defined by all points \((x, y)\) meeting certain criteria such as \(2 \leq x \leq 4\) and \(1 \leq y \leq 5\).

• This ensures an understanding of how sets can describe spatial regions.
• Visualizing these sets helps to comprehend their properties, like boundaries and inclusion.

Open and Closed Sets

The concepts of open and closed sets are fundamental in topology, a branch of mathematics that deals with the properties of space.
An open set is one that does not contain its boundary. Think of it as a region where you can approach its edge as closely as you like, but never quite touch it.
On the other hand, closed sets include all their boundary points. They are "complete" in the sense that they contain everything within and right up to the edges.
In the given problem, the region \( \{ (x, y): 2 \leq x \leq 4, 1 \leq y \leq 5 \} \) is a closed set because it includes its boundary points—those points where \(x\) equals 2 or 4, and where \(y\) equals 1 or 5. This inclusion is confirmed by the less than or equal to (\( \leq \)) inequalities used, meaning all edges are part of the set.

Cartesian Plane

The Cartesian plane is an essential concept in coordinate geometry. It is a two-dimensional plane formed by two perpendicular axes: the horizontal axis (x-axis), and the vertical axis (y-axis). These axes divide the plane into four quadrants.
The Cartesian plane allows us to graphically represent algebraic equations and sets of points. Each point in this plane can be described by an ordered pair \((x, y)\) representing its position along the x and y axes, respectively.
In the context of the exercise given, the Cartesian plane is used to plot the set of points that form the rectangle spanning from \( (2, 1) \) to \( (4, 5) \), and all the interior points within this boundary. This visual representation helps in understanding the nature of the set, its boundary, and determining its openness or closedness.

Boundary of a Set

The boundary of a set plays a critical role in defining its closure properties, distinguishing whether it is open, closed, or neither. In coordinate geometry, the boundary of a set typically includes those points that mark the extent or limit of the set.
The boundary of the given set is the perimeter of the rectangle formed between the points \((2, 1)\) and \((4, 5)\). This includes all points along the sides of the rectangle:

• The line \( x=2 \) from \( y=1 \) to \( y=5 \)
• The line \( x=4 \) from \( y=1 \) to \( y=5 \)
• The line \( y=1 \) from \( x=2 \) to \( x=4 \)
• The line \( y=5 \) from \( x=2 \) to \( x=4 \)

This set is closed because all these boundary points are included. Understanding boundaries is key to classifying sets as open or closed.