Problem 5 - The Calculus with Analytic Geometry

Question

In Exercises 1 through 10 , list the elements of the given set if \(A=\\{0,2,4,6,8\\}, B=\\{1,2,4,8\\}, C=\\{1,3,5,7,9\\}\), and \(D=\) \(\\{0,3,6,9\\}\) $$ B \cup D $$

Step-by-Step Solution

Answer

\( \{0, 1, 2, 3, 4, 6, 8, 9\} \)

Step 1 - Understand the Union Operation (\( \cup \))

The union of two sets, denoted by \( \cup \), includes all unique elements present in either of the two sets.

Step 2 - Identify elements of set \( B \)

The elements of set \( B \) are \( \{1, 2, 4, 8\} \).

Step 3 - Identify elements of set \( D \)

The elements of set \( D \) are \( \{0, 3, 6, 9\} \).

Step 4 - Combine elements and Remove Duplicates

Combine all elements from set \( B \) and set \( D \), and remove any duplicates. This gives \( \{1, 2, 4, 8, 0, 3, 6, 9\} \).

Step 5 - List the Elements of \( B \cup D \)

The elements of \( B \cup D \) are \( \{0, 1, 2, 3, 4, 6, 8, 9\} \).

Key Concepts

Union of Sets

The union of sets is a fundamental operation in set theory. When we perform the union of two sets, we combine all the unique elements from both sets into one set. The symbol used for union is \( \cup \).

For instance, if we have two sets \( A = \{1, 2, 3\} \) and \( B = \{2, 3, 4\} \), the union of these sets, denoted \( A \cup B \), will include all the elements present in both sets without duplication. Thus, \( A \cup B = \{1, 2, 3, 4\} \).

In our exercise, we needed to find the union of sets \( B \) and \( D \). By identifying the elements in each set:

• Set \( B = \{1, 2, 4, 8\} \)
• Set \( D = \{0, 3, 6, 9\} \)

We then combine these elements and remove duplicates to get \( B \cup D = \{0, 1, 2, 3, 4, 6, 8, 9\} \).

Mathematical Sets

A set in mathematics is a collection of distinct objects, considered as an object in its own right. Sets are usually denoted using curly brackets \( \{ \} \) and can contain numbers, symbols, or even other sets.

Here are some key points about sets:

• Sets do not have duplicate elements.
• The order of elements in a set does not matter.

For example, the set \( \{2, 4, 6\} \) is the same as \( \{6, 4, 2\} \).

In the exercise, sets were defined for specific collections of numbers, such as \( B = \{1, 2, 4, 8\} \) and \( D = \{0, 3, 6, 9\} \). Each element in these sets is unique, and they form the basis for performing set operations like union.

Remove Duplicates in Sets

Removing duplicates is an essential aspect of working with sets. Since a set cannot contain duplicate elements, any repeated elements are automatically discarded.

When performing operations like union, combining the elements of two or more sets, we must ensure that any duplicates are removed to maintain the integrity of the set.

In our exercise, after combining elements from sets \( B \) and \( D \):

• Set \( B = \{1, 2, 4, 8\} \)
• Set \( D = \{0, 3, 6, 9\} \)

We combined them to get \( \{1, 2, 4, 8, 0, 3, 6, 9\} \). Since there were no duplicates initially, our final result for \( B \cup D \) was \( \{0, 1, 2, 3, 4, 6, 8, 9\} \), still maintaining no repeated elements.

This process ensures the set remains accurate and follows mathematical principles.