Problem 33 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Let \(U=\\{1,2,3,4,5,6,7,8,9,10\\}\) \(A=\\{1,3,5,7,9\\}, B=\\{2,4,6,8,10\\}\), and \(C=\\{1,2,4,\), \(5,8,9\\}\). List the elements of each set. a. \(A^{c}\) b. \(B \cup C\) c. \(C \cup C^{c}\)

Step-by-Step Solution

Answer

a. \(A^c = \{2, 4, 6, 8, 10\}\) b. \(B \cup C = \{1, 2, 4, 5, 6, 8, 9, 10\}\) c. \(C \cup C^c = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}\)

Step 1: Understand the complement of a set

The complement of a set is the set containing all the elements in the universal set that are not in the given set. To find the complement of a set, simply list all the elements in the universal set that are not in the set you are trying to find the complement of.

Step 2: Find the complement of set A

To find the complement of set A, denoted as \(A^c\), we need to list all the elements in U that are not in A. Since \(A = \{1, 3, 5, 7, 9\}\) and \(U = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}\), we can see that the elements not in A are \(2, 4, 6, 8,\) and \(10\). Thus, the complement of A is: \(A^c = \{2, 4, 6, 8, 10\}\).

Step 3: Understand set union

The union of two sets is the set containing all the elements that are either in one of the sets or in both. To find the union of two sets, list all the elements that are present in either set or both sets.

Step 4: Find the union of sets B and C

To find the union of sets B and C, denoted as \(B \cup C\), we need to list all the unique elements that are either in B or C or both. Since \(B = \{2, 4, 6, 8, 10\}\) and \(C = \{1, 2, 4, 5, 8, 9\}\), we can see that their union contains the elements \(1, 2, 4, 5, 6, 8, 9,\) and \(10\). Thus, the union of B and C is: \(B \cup C = \{1, 2, 4, 5, 6, 8, 9, 10\}\).

Step 5: Find the union of set C and its complement

To find the union of set C and its complement, denoted as \(C \cup C^c\), we first need to find the complement of C. Then, we will list all the unique elements that are present in either set C or its complement or both. Notice that a set's union with its complement results in the universal set. This is because all elements in the universal set are either in the set or in its complement. Therefore, \(C \cup C^c = U = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}\). The results of the exercise are: a. \(A^c = \{2, 4, 6, 8, 10\}\) b. \(B \cup C = \{1, 2, 4, 5, 6, 8, 9, 10\}\) c. \(C \cup C^c = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}\)

Key Concepts

Set Complement: Understanding Complement Sets

In set theory, the concept of the set complement is about identifying what isn't in a given set relative to a larger, encompassing set known as the universal set. The complement of a set, denoted as \(A^c\), includes all elements from the universal set that are not present in set \(A\). This process involves subtracting or removing elements found in \(A\) from those in the universal set \(U\). For example, if \(U = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}\) and \(A = \{1, 3, 5, 7, 9\}\), then the elements \(2, 4, 6, 8,\) and \(10\) make up the complement of \(A\), or \(A^c = \{2, 4, 6, 8, 10\}\). This concept is crucial for understanding operations and relationships between sets as it highlights what is absent as much as what is present.

Set Union: Combining Elements

The union of sets in set theory represents the combination of all distinct elements belonging to either one set or another, without duplication. Using the union operation, symbolized as \(A \cup B\), we form a set containing every element that appears in either set \(A\) or set \(B\), or in both. To illustrate, consider sets \(B = \{2, 4, 6, 8, 10\}\) and \(C = \{1, 2, 4, 5, 8, 9\}\). The union of \(B\) and \(C\) results in \(B \cup C = \{1, 2, 4, 5, 6, 8, 9, 10\}\), encompassing all unique elements from both \(B\) and \(C\). Union is foundational in set theory as it helps in analyzing data combinations and understanding overlapping parts of different datasets.

Universal Set: The Larger Context

A universal set, in set theory, serves as the comprehensive set that contains all relevant elements for a particular discussion or problem. This set is represented as \(U\) and sets the boundary for all other sets under consideration, allowing operations like union and complement to have a frame of reference.When applied to the set operation \(C \cup C^c\), where \(C\) is a subset of \(U\), it exemplifies that a set combined with its complement covers every possible element in the universal set. For instance, \(U = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}\) makes clear that \(C \cup C^c\) will include every element from \(U\), illustrating a primary property of universal sets: they encompass total completion of a set and its complement. Understanding the universal set is essential in set operations to correctly define and understand the scope of element presence and absence in set-based problems.