Question

Describe the following sets as open, closed, or neither, and find \(S^{0},\left(S^{c}\right)^{0},\) and \(\left(S^{0}\right)^{c}\) (a) \(S=(-1,2) \cup[3, \infty)\) (b) \(S=(-\infty, 1) \cup(2, \infty)\) (c) \(S=[-3,-2] \cup[7,8]\) (d) \(S=\\{x \mid x=\) integer \(\\}\)

Short answer

Question: Determine the complement of the interior, types of the sets, and the interiors of the sets, as well as the interior of the complement for each given set: (a) $S=(-1,2)\cup[3,\infty)$, (b) $S=(-\infty,1)\cup(2,\infty)$, (c) $S=[-3,-2]\cup[7,8]$, (d) $S=\{x\mid x=\text{integer}\}$. Answer: (a) Neither open nor closed; interior $(S^0) = (-1,2)\cup(3,\infty)$; interior of the complement $ (S^c)^0 = (2,3)$; complement of the interior $(S^0)^c=(-\infty,-1]\cup[2,3]\cup\{3\}$. (b) Open; interior $(S^0) = (-\infty,1)\cup(2,\infty)$; interior of the complement $ (S^c)^0 = \emptyset$; complement of the interior $(S^0)^c=[1,2]$. (c) Closed; interior $(S^0) = (-3,-2)\cup(7,8)$; interior of the complement $ (S^c)^0 = (-\infty,-3)\cup(-2,7)\cup(8,\infty)$; complement of the interior $(S^0)^c = (-\infty,-3]\cup[-2,7]\cup[8,\infty)$. (d) Neither open nor closed; interior $(S^0) = \emptyset$; interior of the complement $ (S^c)^0 = \bigcup_{n=-\infty}^{\infty}(n,n+1)$; complement of the interior $(S^0)^c = \{x\mid x=\text{integer}\}$.

Step-by-step solution

Determine the type of the set S

The set \(S=(-1,2)\cup[3,\infty)\) consists of two intervals. One is an open interval, and the other is a closed interval. Thus, S is neither open nor closed.

Find the interior of the set S

The interior of S, \(S^0\), consists of all interior points of S. In this case, \(S^0 = (-1,2)\cup(3,\infty)\).

Calculate the interior of the complement of S

The complement of S is \(S^c = [-1,2]\cap(2,3]\). Then, the interior of the complement is: \((S^c)^0 = (2,3)\).

Determine the complement of the interior of S

The complement of the interior of S is found as: \((S^0)^c = (-\infty,-1]\cup[2,3]\cup\{3\}\). (b)

Determine the type of the set S

The set \(S=(-\infty,1)\cup(2,\infty)\) consists of two open intervals. Thus, S is open.

Find the interior of the set S

Since S is an open set, the interior of S is equal to S: \(S^0 = S=(-\infty,1)\cup(2,\infty)\).

Calculate the interior of the complement of S

The complement of S is \(S^c = [1,2]\). The interior of the complement is empty since there are no open intervals: \((S^c)^0 = \emptyset\).

Determine the complement of the interior of S

The complement of the interior of S, \((S^0)^c\), is equal to the complement of S: \((S^0)^c = S^c = [1,2]\). (c)

Determine the type of the set S

The set \(S=[-3,-2]\cup[7,8]\) consists of two closed intervals. Thus, S is closed.

Find the interior of the set S

As S is closed, the interior of S, \(S^0\), consists of the open intervals within S: \(S^0 = (-3,-2)\cup(7,8)\).

Calculate the interior of the complement of S

The complement of S is \(S^c = (-\infty,-3)\cup(-2,7)\cup(8,\infty)\). The interior of the complement is equal to the complement, since it's an open set : \((S^c)^0 = S^c\).

Determine the complement of the interior of S

The complement of the interior of S is found as: \((S^0)^c = (-\infty,-3]\cup[-2,7]\cup[8,\infty)\). (d)

Determine the type of the set S

The set \(S=\{x\mid x=\text{integer}\}\) consists of separate points and does not form any intervals. Thus, S is neither open nor closed.

Find the interior of the set S

As S consists of individual points, it has no interior points, so the interior of S is empty: \(S^0 = \emptyset\).

Calculate the interior of the complement of S

The complement of S is \(S^c = \{x\mid x\neq\text{integer}\}\). The interior of the complement consists of the open intervals that do not contain any integers: \((S^c)^0 = \bigcup_{n=-\infty}^{\infty}(n,n+1)\).

Determine the complement of the interior of S

The complement of the interior of S is equal to S since its interior is empty: \((S^0)^c = S = \{x\mid x=\text{integer}\}\).

Key concepts

Open and Closed Sets

In real analysis, the concepts of open and closed sets are fundamental. An **open set** is a set that does not include its boundary points. Imagine it like a ball without its surface; you can go as close as you want to the edge, but you can't touch it. This means for any point in the set, you can find a small neighborhood around it that stays entirely within the set.
In contrast, a **closed set** includes its boundary points, much like how a closed box includes its edges. If you consider the closure of a set, it adds all the boundary points to that set.

• Open Sets: They exclude boundary points. Intervals like \( (a, b) \) are open because they don't contain \(a\) and \(b\).
• Closed Sets: They include boundary points. Intervals like \[ [a, b] \] are closed because they contain \(a\) and \(b\).
• Neither Open nor Closed: Some sets can be neither, like \( (-1, 2) \cup [3, \infty) \).

Understanding whether a set is open, closed, or neither helps in analyzing its properties and behavior in different contexts.

Set Complement

The concept of a **complement** in set theory is fundamentally about opposites. A complement of a set \( S \), denoted by \( S^c \), consists of all elements not in \( S \). Imagine it like everything outside a circle if \( S \) is the circle.
The complement can change how we understand the original set. When it comes to open and closed sets, the complement also gives insight into the properties of the set.

• If \( S \) is closed, \( S^c \) can be an open set.
• If \( S \) is open, \( S^c \) might be closed or include other elements that are neither.
• The interior of a complement \( (S^c)^0 \) refers to the set of interior points within the complement set, which provide a different perspective.

Working with set complements allows for a deeper understanding of a set's makeup and its relation to the universal set.

Set Interior

The **interior** of a set, denoted \( S^0 \), refers to all points that are not on the edge or boundary. Every point in the interior has a neighborhood wholly contained within the set. This is crucial because it tells us about the core part of the set that does not involve boundary elements.
Finding the interior of different sets can help identify which points are purely within and influential for certain properties of the set.

• For open sets, the interior is the set itself because there are no boundary points to exclude.
• For closed sets, the interior excludes any boundary points, focusing only on internal points.
• If there are no interior points, such as in a set of discrete points, the interior is empty \( \emptyset \).

Understanding the interior of a set gives insights into its structure and helps in formulating proofs and solving problems related to the set. It's important to know how to find it and what it represents for any given set.