Question

Prove: (a) \(\partial\left(S_{1} \cup S_{2}\right) \subset \partial S_{1} \cup \partial S_{2}\) (b) \(\partial\left(S_{1} \cap S_{2}\right) \subset \partial S_{1} \cup \partial S_{2}\) (c) \(\partial \bar{S} \subset \partial S\) (d) \(\partial S=\partial S^{c}\) (e) \(\partial(S-T) \subset \partial S \cup \partial T\)

Short answer

Question: Prove the following relationships between sets S and T: \(\partial(S - T) \subset \partial S \cup \partial T\). Solution: If we assume an arbitrary element \(x \in \partial(S - T)\), this means that for every open ball \(B(x, r)\) centered at x with radius \(r\), the intersection with the set and its complement is non-empty: \(B(x, r) \cap (S - T) \ne \emptyset\) and \(B(x, r) \cap ((S - T)^c) \neq \emptyset\). From this assumption, we can conclude that \(B(x, r) \cap S \ne \emptyset\) and \(B(x, r) \cap (T \cup S^c) \ne \emptyset\). This results in either \(B(x, r) \cap T \ne \emptyset\) or \(B(x, r) \cap S^c \ne \emptyset\). These conditions imply that \(x \in (\partial S \cup \partial T)\), and so, we can conclude that \(\partial(S-T) \subset \partial S \cup \partial T\).

Step-by-step solution

(a) Definition and Goal

We are given two sets, \(S_1\) and \(S_2\), and we are to prove that \(\partial(S_1 \cup S_2) \subset \partial S_1 \cup \partial S_2\). The symbol \(\partial\) denotes the boundary of a set, and the goal is to show that the boundary of the union of the two sets is a subset of the union of the respective boundaries of two sets.

(a) Proof

Assume an arbitrary element \(x \in \partial(S_1 \cup S_2)\). Therefore, for every open ball \(B(x, r)\) centered at x with radius \(r\), the intersection with the set and its complement is non-empty: \(B(x, r) \cap (S_1 \cup S_2) \ne \emptyset\) and \(B(x, r) \cap (S_1^c \cap S_2^c) \neq \emptyset\). Now the intersection of the open ball \(B(x, r)\) with \(S_1\) and \(S_2\) will also be non-empty: \(B(x, r) \cap S_1 \ne \emptyset\) or \(B(x, r) \cap S_2 \ne \emptyset\), and \(B(x, r) \cap S_1^c \ne \emptyset\) or \(B(x, r) \cap S_2^c \ne \emptyset\). This implies that \(x \in \partial S_1 \cup \partial S_2\). Hence, \(\partial(S_1 \cup S_2) \subset \partial S_1 \cup \partial S_2\).

(b) Definition and Goal

We are given two sets, \(S_1\) and \(S_2\), and we are to prove that \(\partial(S_1 \cap S_2) \subset \partial S_1 \cup \partial S_2\). The symbol \(\partial\) denotes the boundary of a set, and the goal is to show that the boundary of the intersection of the two sets is a subset of the union of the respective boundaries of two sets.

(b) Proof

Assume an arbitrary element \(x \in \partial(S_1 \cap S_2)\). Therefore, for every open ball \(B(x, r)\) centered at x with radius \(r\), the intersection with the set and its complement is non-empty: \(B(x, r) \cap (S_1 \cap S_2) \ne \emptyset\) and \(B(x, r) \cap (S_1^c \cup S_2^c) \neq \emptyset\). From this, either \(B(x, r) \cap S_1^c \ne \emptyset\) or \(B(x, r) \cap S_2^c \ne \emptyset\). Also, for the intersection \(B(x, r) \cap (S_1 \cap S_2) \ne \emptyset\), either \(B(x, r) \cap S_1 \ne \emptyset\) or \(B(x, r) \cap S_2 \ne \emptyset\). This implies that \(x \in \partial S_1 \cup \partial S_2\). Hence, \(\partial(S_1 \cap S_2) \subset \partial S_1 \cup \partial S_2\).

(c) Definition and Goal

We have been given a set \(S\), and we are to prove that the boundary of the complement of S, \(\partial \bar{S}\) is a subset of the boundary of S, \(\partial S\).

(c) Proof

This is a direct consequence of the previous problem (d). From (d), we know that \(\partial S=\partial S^{c}\). Therefore, \(\partial \bar{S} \subset \partial S\).

(d) Definition and Goal

We have been given a set \(S\), and the goal is to show that \(\partial S = \partial S^c\).

(d) Proof

Assume an arbitrary element \(x \in \partial S\). This implies that for any open ball \(B(x, r)\) centered at x with radius \(r\), the intersection with the set and its complement is non-empty: \(B(x, r) \cap S \ne \emptyset\) and \(B(x, r) \cap S^c \ne \emptyset\). The same condition holds true if we replace \(S\) with \(S^c\). Hence, \(x \in \partial S^c\) and \(\partial S = \partial S^c\).

(e) Definition and Goal

We are given two sets, \(S\) and \(T\), and we are to prove that \(\partial(S - T) \subset \partial S \cup \partial T\). The goal is to show that the boundary of the set difference is a subset of the union of the respective boundaries of the two sets.

(e) Proof

Assume an arbitrary element \(x \in \partial(S - T)\). Therefore, for every open ball \(B(x, r)\) centered at x with radius \(r\), the intersection with the set and its complement is non-empty: \(B(x, r) \cap (S - T) \ne \emptyset\) and \(B(x, r) \cap ((S - T)^c) \neq \emptyset\). From this, we know that \(B(x, r) \cap S \ne \emptyset\) and \(B(x, r) \cap (T \cup S^c) \ne \emptyset\). So either \(B(x, r) \cap T \ne \emptyset\) or \(B(x, r) \cap S^c \ne \emptyset\). This implies that \(x \in (\partial S \cup \partial T)\). Hence, \(\partial(S-T) \subset \partial S \cup \partial T\).

Key concepts

Set Theory

Set theory is an essential branch of mathematical logic that deals with the nature of sets, which are collections of distinct objects considered as a whole. These objects, called elements or members, can be anything: numbers, people, letters, etc. Set theory provides a foundational system for much of mathematics and includes concepts like subsets, unions, intersections, and complements.

• A **set** is typically denoted by a capital letter and its elements are listed within curly brackets, e.g., \( S = \{ 1, 2, 3 \} \).
• A **union** (\( S_1 \cup S_2 \)) involves all elements present in either set, capturing the idea of combining two sets together.
• An **intersection** (\( S_1 \cap S_2 \)) includes elements that are common to both sets.
• A **complement** (\( S^c \)) contains all elements not in the set, relative to a larger universe of discourse.

To understand how these relate to the problem at hand, consider that the union or intersection impacts the boundaries of the combined sets. This exercise explores how such combinations affect boundaries, a topic deeply rooted in set theory principles.

Boundary of a Set

In real analysis, the boundary of a set \( S \) is a fascinating concept that involves points where the set "meets" its complement. Formally, the boundary \( \partial S \) includes all points such that every open ball centered at that point has some overlap with both the set and its complement.

• This means any point \( x \) in the boundary satisfies: Every neighborhood around \( x \) includes points both in \( S \) and not in \( S \).
• Boundaries can effectively describe the "edge" of a set, important in topology and continuous analysis.

This concept is critical when proving subset relationships involving boundaries. For instance, when focusing on a union \( S_1 \cup S_2 \), its boundary encompasses interactions from each set, stressing why the boundary of the union is a subset of the union of individual boundaries.

Subset Proof

A subset proof is a technique used to demonstrate that every element of one set is also an element of another. In mathematical notation, we say set \( A \) is a subset of set \( B \) if \( A \subset B \).

• To prove \( A \subset B \), we select an arbitrary element of \( A \) and show that this element must also be in \( B \).
• This logical argument underlies many proofs in real analysis, including those involving the boundary of sets. For instance, proving \( \partial(S_1 \cup S_2) \subset \partial S_1 \cup \partial S_2 \) involves selecting a point in the boundary of the union and showing it must belong to one or both set boundaries.

Such proofs require clear and rigorous reasoning to ensure no element is incorrectly excluded or included, reflecting the precision needed in mathematical arguments.

Open Ball Concept

The concept of an open ball is pivotal when analyzing boundaries and exploring real analysis. An open ball \( B(x, r) \) in a metric space is a set of all points that are less than a distance \( r \) from a central point \( x \).

• Mathematically, an open ball is defined as \( B(x, r) = \{ y \in \, \text{space} \, | \, d(x, y) < r \} \), where \( d \) is a distance function like the Euclidean distance.
• Open balls are instrumental in defining the neighborhood of a point, which in turn helps determine if a point is on the boundary of a set.

Recognizing whether two sets coincide in open balls around a point plays a key role in boundary proofs. The idea is simple: manipulate open balls to check interactions with the set and its complement, effectively identifying boundary elements that might not be part of \( S_1 \) or \( S_2 \) but lie in their boundary in a union or intersection.