Question

Sketch the indicated set. Describe the boundary of the set. Finally, state whether the set is open, closed, or neither. \(\\{(x, y): x=0, y=1 / n, n\) a positive integer \(\\}\)

Short answer

The set consists of discrete points along the y-axis; it is neither open nor closed.

Step-by-step solution

Interpret the Set

The given set consists of points (x, y) where the x-coordinate is always 0, and the y-coordinate is of the form \( y = \frac{1}{n} \), where \( n \) is a positive integer. This implies we have points \((0, 1), (0, \frac{1}{2}), (0, \frac{1}{3}), \dots\).

Sketch the Set

To sketch the set, plot each point on the Cartesian plane. The x-coordinate for all these points is 0, so they lie along the y-axis. The plotted points are \((0, 1)\), \((0, \frac{1}{2})\), \((0, \frac{1}{3})\), etc., all aligning vertically on the y-axis.

Describe the Boundary

The boundary of the set is difficult to define in traditional terms since the set consists of distinct points. However, as we consider larger values of \( n \), the points get closer to the x-axis. The density of these points increases without covering the interval completely because no value \( y > 0 \) is actually included as a limit point.

Define Openness or Closedness

A set is closed if it contains all its limit points. Here, the limit as \( n \) approaches infinity is 0 (not included in the set), so it doesn't contain the limit point \( (0, 0) \). A set is open if every point is an interior point, but none of these discrete points are interior. Consequently, the set is neither open nor closed.

Key concepts

Boundary Description

When we talk about the 'boundary' of a set in set theory, we are essentially identifying the 'edges' or 'limits' of that set, where it begins and ends. In this exercise, the set of points given is unique because it consists of individual, separate points along the y-axis of the Cartesian plane.
A boundary here is not so clear-cut. Each point \( (0, \frac{1}{n}) \) doesn't touch the next, because between \( (0, \frac{1}{n}) \) and \( (0, \frac{1}{n+1}) \), there's a continuous space of y-values not included in the set.
📌 **Understanding the Limit:** As \( n \) increases, the y-values get closer and closer to 0. Despite this approach to the x-axis, the actual value of \( y = 0 \) (i.e., the "limit point" \( (0, 0) \)) isn't part of our given set. This absence makes the concept of a boundary for these points a bit different from that of filled areas.
All these points align to suggest that they get infinitely close to the x-axis but never actually form a continuous line along it. A boundary, in this context, is the conceptual edge emerging from this pattern.

Closed and Open Sets

In the realm of set theory, determining whether a set is "closed," "open," or "neither" requires examining the characteristics of limit points and interior points.

• **Closed set**: A set is closed if it includes all its limit points. In this example, the set approaches the limit point \((0, 0)\) as \( n \) becomes infinitely large, but \((0, 0)\) is not part of the set. Therefore, it is not closed.
• **Open set**: A set is open if every point is an interior point. For a point to be interior, you must be able to fit a circle around it where all points in the circle also belong to the set. However, since our points are discrete and entirely isolated, they are not interior points.

Thus, the given set is neither open nor closed, due to the distinct nature of its points that never completely enclose nor isolate the hypothetical limits.

Cartesian Plane Plotting

Plotting points on a Cartesian plane is integral to visualizing mathematical concepts. In this exercise, we look at the specific set of points plotted on the Cartesian plane to understand their arrangement and the implications for understanding set boundaries and openness.
To begin, all points have the x-coordinate 0, meaning they lie exactly on the y-axis. The y-coordinates are fractions getting increasingly smaller, like \((0, 1), (0, \frac{1}{2}), (0, \frac{1}{3}), \ldots\). As \( n \) approaches infinity, the value of \( \frac{1}{n} \) approaches zero, causing the points to move closer to the x-axis.

• ❗ **Key Visualization Tip**: Imagine starting at the point \((0, 1)\) on the y-axis. As you plot each successive point \((0, \frac{1}{2})\), \((0, \frac{1}{3})\), and so on, you move downwards, each step getting closer to the x-axis but never quite reaching it completely.
• Using graph paper or a drawing tool, place a dot for each point and ensure to maintain the declining distance from each successive dot to the origin along the y-axis.

This visual blueprint on the Cartesian plane offers a clear perspective on the continuity or lack thereof in the set's composition and supports the understanding of its open or closed nature.