Problem 24 - Intermediate Algebra

Question

Let \(A=\\{0,1,2,3,4,5,6\\}, B=\\{4,6,8,10\\},\) \(C=\\{-3,-1,0,1,2\\},\) and \(D=\\{-3,1,2,5,8\\}\) Find each set. \(C \cup D\)

Step-by-Step Solution

Answer

The union is \(C \cup D = \{-3, -1, 0, 1, 2, 5, 8\}\).

Step 1: Understand the Union Operation

The union of two sets, denoted as \(C \cup D\), is a set that contains all the elements from both sets, without any duplicates.

Step 2: Write Elements of Set C

List all elements in set C. Given \(C=\{-3,-1,0,1,2\}\), the elements are: -3, -1, 0, 1, and 2.

Step 3: Write Elements of Set D

List all elements in set D. Given \(D=\{-3,1,2,5,8\}\), the elements are: -3, 1, 2, 5, and 8.

Step 4: Combine Elements with No Duplicates

Combine the elements from sets C and D into a single set, removing any duplicate elements. The combined set is: -3, -1, 0, 1, 2, 5, 8.

Step 5: Write the Final Union Set

The set that represents the union of sets C and D is: \(C \cup D = \{-3, -1, 0, 1, 2, 5, 8\}\).

Key Concepts

Union of Sets

The concept of union in set theory is a fundamental tool that allows us to combine elements from two sets. When we refer to the 'union of sets,' we are describing a method to merge sets together, forming one cohesive set. This operation is symbolized with a union symbol \( \cup \).

Let's consider two sets, just like in the exercise, set \( C \) and set \( D \). The union \( C \cup D \) consists of all unique elements that are present in either set \( C \) or set \( D \). Importantly, if an element appears in both sets, it is still represented only once in the resulting union set.

The union operation is widely used as it helps to collate and organize information, making it easier to manage and understand large data sets by viewing them as a single entity.

To perform a union operation, simply list all elements from both sets, ensuring no duplicates, to obtain the final union set.

Duplicate Elements

Understanding how to deal with duplicate elements is crucial when working with the union of sets. When two sets are combined, any element that appears more than once is only included once in the resulting set. This characteristic of sets is important in distinguishing them from lists or multi-sets where duplication can exist freely.

Consider the sets from the exercise:

• Set \( C = \{-3, -1, 0, 1, 2\} \)
• Set \( D = \{-3, 1, 2, 5, 8\} \)

When forming the union \( C \cup D \), we note that the elements \(-3, 1,\) and \(2\) appear in both sets. Thus, each of these elements is included only once in the final result, as set theory requires a unique representation for each element.

This helps in maintaining concise and clear representation of data when dealing with large sets, avoiding redundancy and promoting efficiency.

Mathematical Notation

The use of mathematical notation is an efficient way to communicate and solve problems involving sets. Notation provides a universal language which simplifies the representation and understanding of mathematical concepts.

In our exercise, the notation \( C \cup D \) succinctly expresses the concept of the union between two sets. This notation ensures that mathematicians and students all over the world can understand that we are discussing all elements present in either set \( C \) or \( D \), without any need for additional explanation by showing:

• \( C \) is \( \{-3, -1, 0, 1, 2\} \)
• \( D \) is \( \{-3, 1, 2, 5, 8\} \)
• \( C \cup D \) results in \( \{-3, -1, 0, 1, 2, 5, 8\} \)

Mathematical notation thus becomes a crucial part of understanding and teaching set operations, making complicated operations understandable and manageable.