Problem 10 - Advanced Calculus

Question

Construct pictures to show that each of the following is false. (i) If \(A \subset B\). then \(\operatorname{bdy}(A) \subset \operatorname{bdy}(B)\) (ii) \(\operatorname{bd} y(S)\) is the same as the boundary of the closure of \(S\). (iii) \(\operatorname{bdy}(S)\) is the same as the boundary of the interior of \(S\). (iv) The interior of \(S\) is the same as the interior of the closure of \(S .\)

Step-by-Step Solution

Answer

All of these statements are false.

Step 1: Disprove (i) If \(A \subset B\), then \(\operatorname{bdy}(A) \subset \operatorname{bdy}(B)\)

Let A be an open set contained in the closed set B such that A doesn't share any boundary points with B. In such a case \(\operatorname{bdy}(A)\) would not be a subset of \(\operatorname{bdy}(B)\) which contradicts the given statement.

Step 2: Disprove (ii) \(\operatorname{bd} y(S)\) is the same as the boundary of the closure of \(S\)

For the set \(S = (0,1)\) (an open interval), boundary of \(S\) is {0,1}. But when we take closure of \(S\) which is [0,1], the boundary remains the same. This contradicts the statement.

Step 3: Disprove (iii) \(\operatorname{bdy}(S)\) is the same as the boundary of the interior of \(S\)

Let \(S = [0,1]\) which is a closed set. The boundary of \(S\) is {0,1}. While the interior of \(S\) is (0,1) whose boundary is also {0,1}. This contradicts the statement.

Step 4: Disprove (iv) The interior of \(S\) is the same as the interior of the closure of \(S \)

Let \(S = (0, 1)\). The interior of \(S\) is again (0,1). However, the closure of \(S\) is [0,1] and its interior is again (0,1). This disapproves the statement.

Key Concepts

Set Theory

Set theory is the mathematical study of collections of objects, known as sets. In basic terms, a set is a grouping of distinct objects, called elements or members, without any particular order. Sets can be anything from a collection of numbers to a range of concepts. Understanding the properties of sets and the relationships between them is essential in mathematics.

For instance, when we define a set named 'A' that includes the numbers 1, 2 and 3, we write it as A = {1, 2, 3}. Set theory lies at the foundation of various mathematical disciplines including algebra, geometry, and calculus, and has applications in logic and computer science as well. One key aspect of set theory is how sets interact with each other which includes operations like union, intersection, and difference, as well as concepts like subsets, where one set is entirely contained within another.

Boundary (bdy) of a Set

In mathematics, the boundary (bdy) of a set is a fundamental concept that delves into the 'edges' or 'limits' of that set. The boundary of a set 'A' consists of all points that can be approached from both inside and outside of 'A'. In other words, it's the set of points that are neither completely encompassed by the interior nor the exterior of 'A'.

For a more concrete visualization, consider a circle on a plane. The boundary of this set is the circumference of the circle, whereas the interior is the space within, excluding the circumference. This concept is crucial when differentiating between closed sets, which include their boundary points, and open sets, which do not. The exercises provided aim to clarify misconceptions about boundaries of sets and their relationships with other sets, emphasizing the importance of visual examples and counterexamples.

Interior and Closure of Sets

The concepts of interior and closure are integral to understanding the structure of sets. The interior of a set consists of all the points that are not on the boundary; it's the 'inside' part of the set where if you pick any point, you can move a little in any direction and still remain inside the set. For example, in the set of all points inside a rectangle (excluding the boundary), the interior is the space occupied by the rectangle minus its edges.

The closure of a set, on the other hand, is the complete 'encapsulation' of that set, including all the boundary points. It's like sealing the set within an envelope without leaving any gaps. Thus, the closure of a set is always a closed set itself. Understanding these two concepts allows students to discern different types of sets and comprehend exercises concerning boundary points and how they relate to the interior and closure of sets.

Counterexamples in Set Theory

Counterexamples are a powerful tool in mathematics, particularly in set theory, to disprove statements or conjectures. By finding just one instance where a statement doesn't hold, it's proven false. This is crucial for understanding the limitations and conditions that apply to certain theories within set theory.

For example, providing counterexamples can clearly demonstrate that the properties one might assume about sets like the relationship between boundaries of different sets, or the equivalence of interiors and closures do not always hold true. Using specific, often simple, examples helps students grasp why a general statement may be incorrect. This method of disproving by example not only reinforces the nuance in mathematical theorems and properties but also encourages critical thinking and a deeper inquiry into the nature of sets and their boundaries.