Problem 29

Question

Sketch the indicated set. Describe the boundary of the set. Finally, state whether the set is open, closed, or neither. \(\left\\{(x, y): 0

Step-by-Step Solution

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Answer

The set is neither open nor closed, includes the boundary of a unit circle except the origin.

1Step 1: Identify the Set

The set given is $ \{(x, y) : 0 < x^2 + y^2 \leq 1\} $. This describes all points in the coordinate plane such that the distance from the origin, squared, is greater than 0 and less than or equal to 1.

2Step 2: Understand the Boundary

The expression $ x^2 + y^2 = 1 $ describes a circle with radius 1 centered at the origin. The inequality $ 0 < x^2 + y^2 \leq 1 $ indicates the set includes the region inside this circle and also includes its boundary (because it is "less than or equal"). However, points with $ x^2 + y^2 = 0 $, which is just the origin, are excluded.

3Step 3: Sketch the Set

To sketch the set, draw a circle with radius 1 centered at the origin; this represents the boundary $ x^2 + y^2 = 1 $. The shaded area will include all points inside and on the circle except the origin point (0,0), which would be excluded.

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

A set is open if it doesn't include its boundary, and closed if it includes its boundary and is complete. Here, the set includes the boundary described by $ x^2 + y^2 = 1 $, but not its entire boundary as $ x^2 + y^2 = 0 $ is excluded. Hence, the set is neither open nor closed.

Key Concepts

Coordinate GeometryOpen and Closed SetsInequalities in CalculusBoundary of a Set

Coordinate Geometry

Coordinate geometry, often called analytic geometry, connects algebra and geometry through a coordinate system. In a coordinate plane, every point can be represented using a pair of numbers, typically $(x, y)$.To understand sets and regions, it is important to visualize them on this coordinate plane.

The equation $x^2 + y^2 = 1$ represents a circle centered at the origin with a radius of 1.
This is because every point $(x, y)$ that is 1 unit away from the origin satisfies this equation.

In this exercise, we're dealing with inequalities in coordinate geometry, which add a further layer of complexity. The inequality $0 < x^2 + y^2 \leq 1$ indicates we're looking at the region inside the circle but excluding the origin itself. This helps us to define and understand the specific area, or set, that we are focusing on within the problem.

Open and Closed Sets

In the context of set theory and calculus, distinguishing between open and closed sets involves understanding their boundaries:

An open set does not include its boundary.
A closed set includes its boundary and all its limit points.

In the given exercise, the set includes points such that $0 < x^2 + y^2 \leq 1$,which means the circular boundary, $x^2 + y^2 = 1$, is part of the set.
However, the point $(0, 0)$,where $x^2 + y^2 = 0$,is not included in the set, indicating the set is neither fully open nor fully closed. It lacks some parts of what would make it a closed set.

Inequalities in Calculus

Understanding inequalities in calculus is crucial for describing regions on a coordinate plane. Inequalities help in determining which part of a plane a set of points belong to.

The inequality $x^2 + y^2 \leq 1$defines a circle including its interior and boundary.
The restriction $0 < x^2 + y^2$ensures that the smallest circle, merely a point at the origin, is excluded.

Such inequalities refine our understanding of the location and properties of sets within the geometric space, allowing for precise definitions of areas, volumes, and other geometric properties. Inequalities not only help frame these spaces but also underpin solutions in optimization and analysis.

Boundary of a Set

The boundary of a set plays a crucial role in understanding its characteristics in set theory and calculus. The boundary consists of those points that define the limits of the set.

For $x^2 + y^2 \leq 1$,the boundary is the circle defined by $x^2 + y^2 = 1$.
This circle is included in the set, making it part of the boundary comprised of points on the edge of the circle.
However, the exclusion of the origin point,$(0,0)$,from the set means the boundary is incomplete—hence, impacting whether the set can be considered closed or open.

By examining which points form the boundary and which do not, we grasp the complete geometric and topological structure of the set.

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