Problem 4 - Applied Discrete Structures

Question

Let \(U=\\{1,2,3, \ldots, 9\\} .\) Give examples to illustrate the following facts: (a) If \(A \subseteq B\) and \(B \subseteq C,\) then \(A \subseteq C\). (b) There are sets \(A\) and \(B\) such that \(A-B \neq B-A\) (c) If \(U=A \cup B\) and \(A \cap B=\emptyset\), it always follows that \(A=U-B\).

Step-by-Step Solution

Answer

(a) Sets: \(A=\{1,2\}\), \(B=\{1,2,3,4\}\), \(C=\{1,2,3,4,5,6\}\). (b) Sets: \(A=\{1,2\}\), \(B=\{2,3\}\). (c) Sets: \(A=\{1,2,3,4,5\}\), \(B=\{6,7,8,9\}\), verify \(A=U-B\).

Step 1: Understanding the Problem

We need to illustrate three distinct facts about set theory using examples and the universal set \( U = \{1, 2, 3, \ldots, 9\} \). Each point addresses a different relationship or property of sets.

Step 2: Example for Part (a)

Let's choose sets such that \( A = \{1, 2\} \), \( B = \{1, 2, 3, 4\} \), and \( C = \{1, 2, 3, 4, 5, 6\} \). Clearly, \( A \subseteq B \) because all elements of \( A \) are in \( B \), and \( B \subseteq C \) for the same reason. Therefore, \( A \subseteq C \) is true, as all elements of \( A \) are also in \( C \), demonstrating the transitive property of subsets.

Step 3: Example for Part (b)

Define sets \( A = \{1, 2\} \) and \( B = \{2, 3\} \). Calculate \( A - B \) and \( B - A \): \( A - B = \{1\} \) because 1 is in \( A \) and not in \( B \); \( B - A = \{3\} \) because 3 is in \( B \) and not in \( A \). Here, \( A - B eq B - A \) as \( \{1\} eq \{3\} \).

Step 4: Example for Part (c)

Let \( A = \{1, 2, 3, 4, 5\} \) and \( B = \{6, 7, 8, 9\} \), ensuring \( A \cap B = \emptyset \). Since \( U = A \cup B \), then \( U = \{1,2,3,4,5,6,7,8,9\} \). We need to show that \( A = U - B \). Compute \( U - B = \{1, 2, 3, 4, 5\} \), which matches \( A \). Therefore, \( A = U - B \), as the sets perfectly partition \( U \) without overlap.

Key Concepts

universal set

In set theory, the universal set is a comprehensive set that contains all the objects under consideration, typically denoted by the symbol \( U \). All other sets in a particular context are considered subsets of this universal set. For example, if we let \( U = \{1, 2, 3, \ldots, 9\} \), then any set we discuss—such as \( A = \{1, 2, 3\} \) or \( B = \{4, 5\} \)—will be a subset of \( U \). The universal set is crucial because it provides a complete frame of reference for discussing relationships and operations among different sets.
It allows us to perform set operations, such as unions and intersections, knowing that all potential elements are within \( U \). In the exercise, the universal set \( U \) makes it easier to demonstrate and illustrate the properties and relationships of sets \( A \), \( B \), and \( C \).

set operations

Set operations are the mathematical processes applied to sets to combine, relate, and modify them. The primary set operations include:

• Union (\( \cup \)): Combines all elements from two sets. For example, \( A \cup B \) results in a set containing all elements from both \( A \) and \( B \).
• Intersection (\( \cap \)): Produces a set containing only elements present in both sets. If \( A \cap B = \emptyset \), it means the sets have no elements in common.
• Difference (\( - \)): Consists of elements in one set but not in the other. For instance, \( A - B \) includes elements in \( A \) that are not in \( B \).

These operations allow us to explore how sets relate to each other and how they can be transformed. In our examples, these operations show cases like "difference" in part (b), where \( A - B eq B - A \), and the idea of partitioning a universal set using union and differences, as in part (c). Set operations provide a versatile language to express these relationships.

subsets

A subset is a set wherein all elements are also elements of another set. We denote \( A \subseteq B \) to show that \( A \) is a subset of \( B \), meaning every element of \( A \) is found in \( B \). This concept helps determine the inclusivity of one set within another.
Consider the example \( A = \{1, 2\} \), \( B = \{1, 2, 3, 4\} \), and \( C = \{1, 2, 3, 4, 5, 6\} \). We observe:

• \( A \subseteq B \) because 1 and 2 are in \( B \), and
• \( B \subseteq C \) because all elements in \( B \) are also in \( C \).

This demonstrates a chain in which \( A \) is also a subset of \( C \)—illustrating subset relationships beyond immediate neighbors, which we'll elaborate on with the transitive property of subsets.

transitive property of subsets

The transitive property of subsets is a fundamental principle in set theory. It states: if \( A \subseteq B \) and \( B \subseteq C \), then \( A \subseteq C \).
This property asserts the inherent guarantee about the relationship of sets when placed in a sequence. In simpler terms, if \( A \) is inside \( B \), and \( B \) is inside \( C \), then naturally, \( A \) is inside \( C \) as well. The sets from our earlier step-by-step problem illustrate this property perfectly. With \( A = \{1, 2\} \), \( B = \{1, 2, 3, 4\} \), and \( C = \{1, 2, 3, 4, 5, 6\} \):

• Since \( A \subseteq B \), every element of \( A \) is in \( B \).
• And since \( B \subseteq C \), every element of \( B \) must also be in \( C \).

Thus, \( A \subseteq C \), showing that every element of \( A \) is assuredly found in \( C \). This property simplifies the understanding of layered or nested set relationships and is widely applicable in various mathematical scenarios.