Solved Worksheet 2 Sets - Set Operations 1. Let U = {1,2, | Chegg.com - Free Printable

Problem Analysis

We are given the universal set \( U = \{1, 2, 3, 4, 5, 6, 7, 8, 9\} \) and several subsets:
- \( A = \{1, 2, 3, 4, 5\} \)
- \( B = \{4, 5, 6, 7\} \)
- \( C = \{5, 6, 7, 8, 9\} \)
- \( D = \{1, 3, 5, 7, 9\} \)
- \( F = \{1, 5, 9\} \)

We need to find:
1. \( A \cup B \) and \( A \cap B \)
2. \( A \cup C \) and \( A \cap C \)
3. \( D \cup F \) and \( D \cap F \)

Solution

#### Part (a): \( A \cup B \) and \( A \cap B \)

1. Union \( A \cup B \):
- The union of two sets \( A \) and \( B \) is the set of all elements that are in \( A \), in \( B \), or in both.
- \( A = \{1, 2, 3, 4, 5\} \)
- \( B = \{4, 5, 6, 7\} \)
- Combining all unique elements from \( A \) and \( B \):
\[
A \cup B = \{1, 2, 3, 4, 5, 6, 7\}
\]

2. Intersection \( A \cap B \):
- The intersection of two sets \( A \) and \( B \) is the set of all elements that are in both \( A \) and \( B \).
- \( A = \{1, 2, 3, 4, 5\} \)
- \( B = \{4, 5, 6, 7\} \)
- The common elements between \( A \) and \( B \) are:
\[
A \cap B = \{4, 5\}
\]

#### Part (b): \( A \cup C \) and \( A \cap C \)

1. Union \( A \cup C \):
- \( A = \{1, 2, 3, 4, 5\} \)
- \( C = \{5, 6, 7, 8, 9\} \)
- Combining all unique elements from \( A \) and \( C \):
\[
A \cup C = \{1, 2, 3, 4, 5, 6, 7, 8, 9\}
\]

2. Intersection \( A \cap C \):
- \( A = \{1, 2, 3, 4, 5\} \)
- \( C = \{5, 6, 7, 8, 9\} \)
- The common elements between \( A \) and \( C \) are:
\[
A \cap C = \{5\}
\]

#### Part (c): \( D \cup F \) and \( D \cap F \)

1. Union \( D \cup F \):
- \( D = \{1, 3, 5, 7, 9\} \)
- \( F = \{1, 5, 9\} \)
- Combining all unique elements from \( D \) and \( F \):
\[
D \cup F = \{1, 3, 5, 7, 9\}
\]

2. Intersection \( D \cap F \):
- \( D = \{1, 3, 5, 7, 9\} \)
- \( F = \{1, 5, 9\} \)
- The common elements between \( D \) and \( F \) are:
\[
D \cap F = \{1, 5, 9\}
\]

Final Answers

\[
\boxed{
\begin{aligned}
&\text{(a)} \quad A \cup B = \{1, 2, 3, 4, 5, 6, 7\}, \quad A \cap B = \{4, 5\} \\
&\text{(b)} \quad A \cup C = \{1, 2, 3, 4, 5, 6, 7, 8, 9\}, \quad A \cap C = \{5\} \\
&\text{(c)} \quad D \cup F = \{1, 3, 5, 7, 9\}, \quad D \cap F = \{1, 5, 9\}
\end{aligned}
}
\]