Problem 28 - 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. \(\left\\{(x, y): x^{2}+y^{2}<4\right\\}\)

Step-by-Step Solution

Answer

The set is the interior of a circle with radius 2, boundary excluded, and it is open.

Step 1: Identify the Set

The given set is the collection of all points \((x, y)\) such that \(x^2 + y^2 < 4\). This defines a region within the Cartesian plane.

Step 2: Understand the Boundary

The inequality \(x^2 + y^2 < 4\) means we are looking at points strictly inside a circle. The equation \(x^2 + y^2 = 4\) defines the boundary, which is a circle with center at \((0, 0)\) and radius 2.

Step 3: Visualize the Sketch

Sketch the circle defined by \(x^2 + y^2 = 4\). This circle is not part of the set since we only include points where \(x^2 + y^2 < 4\). Shade the interior of the circle, excluding the boundary itself.

Step 4: Determine Openness or Closedness

A set is open if it does not include its boundary and closed if it contains its boundary. The set is defined by \(x^2 + y^2 < 4\), meaning it does not include points on the circle itself. Therefore, this set is open.

Key Concepts

Sketching Mathematical Sets

When sketching mathematical sets, it is essential to translate mathematical expressions into visual representations. This helps in understanding the region or points being described. For instance, the set \(\{(x, y) : x^2 + y^2 < 4\}\) represents a collection of points inside a circle. Here’s how you can sketch it:

• First, identify the boundary by setting the expression to an equality. In this case, \(x^2 + y^2 = 4\) gives us a circle.
• The center of this circle is at the origin \( (0,0) \), and it has a radius of 2.
• To sketch, draw this circle lightly with a dashed line to indicate it is not part of the set.
• Shade the inside of the circle, which represents all points where \(x^2 + y^2 < 4\).

Using a visual approach aids in better comprehension, especially when determining regions that contain or exclude boundaries.

Boundary of a Set

The boundary of a set in a mathematical context refers to the dividing line or surface between the elements that are included in the set and those that are not. For the set \(\{(x, y) : x^2 + y^2 < 4\}\), this boundary is the circumference of the circle defined by \(x^2 + y^2 = 4\).Understanding boundaries is crucial because:

• It helps in identifying if certain elements are part of the set.
• Provides insight into set classification, such as open or closed.

Boundaries can be considered as a clear line that defines inclusion or exclusion, allowing you to determine which points fall within the set and which do not. In this case, the circle's edge is the boundary, but it isn't included in the set itself.

Open and Closed Sets

In topology, sets can be classified as open, closed, or neither. The distinction depends on whether the boundary is included in the set.

• An open set does not include its boundary. In our example of \(x^2 + y^2 < 4\), the set is open because it consists of points inside the circle but not on it.
• A closed set includes its boundary. If the original problem had been expressed as \(x^2 + y^2 \leq 4\), it would be closed.
• A set is neither open nor closed if it partially includes parts of its boundary.

Knowing whether a set is open, closed, or neither is integral for mathematical reasoning and further applications in calculus and analysis, as it influences continuity and limits.