Understanding Set Operations

In set theory, we work with collections of distinct objects called sets. The question asks us to find the complement of the union of two sets, \( A \) and \( B \), with respect to a universal set \( U \).

We are given the following sets:

• Universal Set \( U = \{1, 2, 3, 4, 5, 6, 7, 8, 9\} \)
• Set \( A = \{1, 2, 3, 4\} \)
• Set \( B = \{2, 4, 6, 8\} \)

Calculating the Union of Sets A and B

The union of two sets, denoted by \( A \cup B \), is the set containing all the elements that are in set \( A \) or in set \( B \) or in both.

To find \( A \cup B \), we combine the elements from set \( A \) and set \( B \), listing each element only once.

\( A = \{1, 2, 3, 4\} \)

\( B = \{2, 4, 6, 8\} \)

\( A \cup B = \{1, 2, 3, 4\} \cup \{2, 4, 6, 8\} \)

The elements in \( A \) are 1, 2, 3, 4. The elements in \( B \) are 2, 4, 6, 8. Combining these unique elements gives us:

\( A \cup B = \{1, 2, 3, 4, 6, 8\} \)

Calculating the Complement of the Union

The complement of a set \( S \), denoted by \( S' \) or \( S^c \), with respect to a universal set \( U \), is the set of all elements in \( U \) that are not in \( S \). In this problem, we need to find \( (A \cup B)' \). This means we need to find all elements in the universal set \( U \) that are not in the set \( A \cup B \).

We have:

• Universal Set \( U = \{1, 2, 3, 4, 5, 6, 7, 8, 9\} \)
• Set \( A \cup B = \{1, 2, 3, 4, 6, 8\} \)

To find \( (A \cup B)' \), we look at the elements in \( U \) and remove the elements that are also in \( A \cup B \).

Elements in \( U \) are: 1, 2, 3, 4, 5, 6, 7, 8, 9.

Elements in \( A \cup B \) are: 1, 2, 3, 4, 6, 8.

Removing the elements {1, 2, 3, 4, 6, 8} from {1, 2, 3, 4, 5, 6, 7, 8, 9}, we are left with:

\( (A \cup B)' = \{5, 7, 9\} \)

Final Result

The complement of the union of sets A and B, \( (A \cup B)' \), is the set containing the elements 5, 7, and 9.