Problem 10 - APEX Calculus

Question

A set \(S\) is given. (a) Give one boundary point and one interior point, when possible, of \(S\). (b) State whether \(S\) is open, closed, or neither. (c) State whether \(S\) is bounded or unbounded. $$ S=\\{(x, y) \mid y>\sin x\\} $$

Step-by-Step Solution

Answer

(a) Boundary example: \((x, \sin x)\); Interior example: \((0, 2)\). (b) Open. (c) Unbounded.

Step 1: Identify Boundary Points

Boundary points are those where every neighborhood contains at least one point from the set and one point not in the set. For the set \( S = \{(x, y) \mid y > \sin x\} \), points like \((x, y) = (x, \sin x)\) are on the boundary since any infinitely small circle around \( (x, \sin x) \) contains points both in \( S \) (like \( y = \sin x + 0.1 \)) and not in \( S \) (like \( y = \sin x - 0.1 \)). Thus, any point \( (x, \sin x) \) is a boundary point.

Step 2: Identify Interior Points

Interior points are those where some neighborhood entirely lies within the set. For the set \( S \), if \( y > \sin x + \epsilon \) for some \( \epsilon > 0 \), this point is interior because you can find a small circle around it that lies entirely within \( S \). An example is the point \( (x, y) = (0, 2) \) because if \( r < 2 - \sin 0 \), there is a circle entirely in \( S \). Thus, \( (0, 2) \) can be considered an interior point.

Step 3: Determine if Set is Open or Closed

A set is open if it contains all its interior points and does not contain its boundary points. It is closed if it contains all boundary points and none outside. Since \( S \) does not include points where \( y = \sin x \) (its boundary), it is not closed. However, \( S \) contains all its interior points. Thus, \( S \) is an open set.

Step 4: Determine if Set is Bounded or Unbounded

A set is bounded if all points lie within some finite distance around a central point. The set \( S = \{(x, y) \mid y > \sin x\} \) continues indefinitely in the \( y \)-direction and also along the \( x \)-axis without bounds. Thus, \( S \) is unbounded.

Key Concepts

boundary points

In set theory, boundary points are critical for understanding the edges of a set. A boundary point is a point where every neighborhood around it contains both points from inside the set and points from outside the set. For example, consider the set \( S = \{(x, y) \mid y > \sin x\} \). At any of its boundary points \( (x, \sin x) \), tiny circles encompass points above the sine curve (inside \( S \)) and points on the sine curve (not in \( S \)).

• Boundary points help define the "edge" of a set.
• They provide insight into the set's interactions with its surroundings.

Remember, a boundary does not necessarily belong to the set itself; it's about where it is undeniably close to being inside and outside the set in every small vicinity.

interior points

Interior points are all about being purely inside the set, with some wiggle room. For a point to be considered an interior point, there must exist a small neighborhood around it that fits entirely within the set. Think of it as having a comfortable space where you belong, and no part of that space sneaks out of the set.

• An example for \( S = \{(x, y) \mid y > \sin x\} \) is the point \( (0, 2) \).
• From this point, you can draw a small circle (like \( r < 2 - \sin 0 \)) contained entirely in \( S \).

Interior points are straightforward: they signify positions comfortably surrounded by the set’s elements without touching the boundaries or exterior spaces.

open and closed sets

Sets can be open, closed, both, or neither, depending on their relation to boundary and interior points. An open set boasts all of its interior points but leaves out boundary points. This gives it a feeling of being unrestricted, much like an open-ended group of friends.

• For the set \( S = \{(x, y) \mid y > \sin x\} \), this means including every \( y > \sin x \) without touching \( y = \sin x \), making it open.
• A closed set, conversely, includes all boundary points but none from outside the boundary.

Understanding whether a set is open or closed is fundamental, as it tells us how the set interacts with its boundary and affects the properties and behaviors within more complex structures.

bounded and unbounded sets

The concept of bounded and unbounded sets deal with the size limits of a set. A set is bounded if you can "draw a circle" around it, meaning all points lie within some finite distance from a central point.

• Unbounded sets, like our set \( S = \{(x, y) \mid y > \sin x\} \), stretch on indefinitely.
• In this case, \( S \) extends without any limit in the \( y \)-direction and along the entire \( x \)-axis.

This means there’s no way to encapsulate the entire set within any finite boundary. Knowing whether a set is bounded or unbounded provides insight into its scope and potential effects on the analysis within mathematical or real-world frameworks.