Problem 27 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 27-32, sketch the indicated set. Describe the boundary of the set. Finally, state whether the set is open, closed, or neither. 27\. \(\\{(x, y): 2 \leq x \leq 4,1 \leq y \leq 5\\}\)

Step-by-Step Solution

Answer

The set is a closed rectangle with boundaries at \(x = 2, x = 4, y = 1, y = 5\). Its boundary is its perimeter, and it is closed.

Step 1: Understanding the Set

The given set \(\{(x, y): 2 \leq x \leq 4, 1 \leq y \leq 5\}\) consists of all points \((x, y)\) such that both \(x\) and \(y\) lie within specific intervals. Here, \(x\) ranges from \(2\) to \(4\), and \(y\) ranges from \(1\) to \(5\). This forms a rectangular region on the xy-plane.

Step 2: Sketching the Set

To sketch this set, plot the points where \(x = 2\), \(x = 4\), \(y = 1\), and \(y = 5\). Connect these points to form a rectangle. The vertices of this rectangle will be \((2, 1)\), \((2, 5)\), \((4, 1)\), and \((4, 5)\). The entire shaded area within this rectangle represents the set.

Step 3: Describing the Boundary

The boundary of the set is the perimeter of the rectangle. This is composed of the line segments joining the vertices \((2, 1)\) to \((2, 5)\), \((2, 5)\) to \((4, 5)\), \((4, 5)\) to \((4, 1)\), and \((4, 1)\) to \((2, 1)\). The boundary includes these line segments as the inequalities \(2 \leq x \leq 4\) and \(1 \leq y \leq 5\) include the equal sign.

Step 4: Determining If the Set is Open, Closed, or Neither

The set includes its boundary, as the inequalities are 'less than or equal to.' Therefore, the set is closed. A set is closed if it contains all its boundary points, which this set does by including the equalities \(x = 2\), \(x = 4\), \(y = 1\), and \(y = 5\).

Key Concepts

Rectangular Region

A rectangular region in the coordinate plane is formed by all the points bounded by specific intervals for the x and y coordinates. In the given exercise, the set \(\{(x, y): 2 \leq x \leq 4, 1 \leq y \leq 5\}\) defines such a region. This set implies that both variables, x and y, exist within closed intervals: x between 2 and 4, and y between 1 and 5. Together, these constraints frame a rectangle on the xy-plane.

• The sides of this rectangle are parallel to the coordinate axes.
• Its vertices can be explicitly defined as the points (2, 1), (2, 5), (4, 1), and (4, 5).

Each point inside the rectangle, as well as the points on the boundary, satisfy the given conditions for x and y. Thus, understanding rectangular regions involves identifying these intervals and recognizing the space they create on the plane.

Open and Closed Sets

In set theory, understanding whether a set is open, closed, or neither is key. Let's dive into what "open" and "closed" mean primarily in the context of this problem. An open set doesn't include its boundary points, while a closed set does include them. The set given, \(\{(x, y): 2 \leq x \leq 4, 1 \leq y \leq 5\}\), is a closed set.

• Closed sets: Include every point on the boundary.
• Open sets: Exclude all boundary points.

In this case, because of the "less than or equal to" inequalities in the definition of the set, all boundary points, such as where x = 2 or y = 5, are included. It’s important to note that if either inequality didn’t have the "equal" part, the set would not include those boundary points, making it open or possibly neither open nor closed if some areas were bounded and some were not. Understanding these distinctions helps in determining the properties of a set.

Inequalities

Inequalities are mathematical expressions used to model regions and define boundaries in geometry. In the context of the given set, inequalities are crucial in defining which points lie within the rectangular region. For example, \(2 \leq x \leq 4\) specifies the horizontal constraints, meaning that any point x has to be between 2 and 4, inclusive. Similarly, \(1 \leq y \leq 5\) sets the vertical range for y between 1 and 5, also inclusive.

• "Less than or equal to" (≤) indicates inclusion, meaning boundary points are part of the set.
• "Less than" (<) would indicate exclusion, meaning boundary points are not part of the set.

By applying these inequalities, we can graphically represent a space that encompasses all points satifying these conditions. These mathematical tools allow us to construct specific areas and shapes, like the rectangle discussed, directly in a coordinate system.