Question

What is an example of a question and its solution on how to find the complement of a set given a universal set U?

Answer 1

Certainly! Here’s an example question along with its solution on how to find the complement of a set given a universal set \( U \).

### Example Question:

Let the universal set \( U \) be defined as follows:

\[ U = \{ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 \} \]

Let set \( A \) be defined as:

\[ A = \{ 2, 4, 6, 8 \} \]

What is the complement of set \( A \) in the universal set \( U \)?

### Solution:

The complement of a set \( A \) in a universal set \( U \), denoted as \( A' \) or \( U - A \), consists of all the elements in \( U \) that are not in \( A \).

1. **Identify the elements in the universal set \( U \)**:
\[
U = \{ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 \}
\]

2. **Identify the elements in set \( A \)**:
\[
A = \{ 2, 4, 6, 8 \}
\]

3. **List the elements in \( U \) that are not in \( A \)**:
- The elements in \( U \) are: \( 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 \)
- The elements in \( A \) are: \( 2, 4, 6, 8 \)
- The elements that are **not** in \( A \) are: \( 1, 3, 5, 7, 9, 10 \)

4. **Write the complement of set \( A \)**:
\[
A' = U - A = \{ 1, 3, 5, 7, 9, 10 \}
\]

### Final Answer:
The complement of set \( A \) in the universal set \( U \) is:
\[
A' = \{ 1, 3, 5, 7, 9, 10 \}
\]