Question

If \( U = \{1, 2, 3, 4, 5, 6, 7, 8, 9\} \), \( A = \{2, 4, 6, 8\} \), and \( B = \{2, 3, 5, 7\} \), verify that \( (A \cup B)' = A' \cap B' \).

Hint:

The complement of the union of two sets is the intersection of their complements, i.e., \( (A \cup B)' = A' \cap B' \).

Answer 1

Step 1: Define the sets.
We are given: \[ U = \{1, 2, 3, 4, 5, 6, 7, 8, 9\}, \quad A = \{2, 4, 6, 8\}, \quad B = \{2, 3, 5, 7\} \]
Step 2: Find \( A \cup B \) and its complement.
The union of \( A \) and \( B \) is: \[ A \cup B = \{2, 3, 4, 5, 6, 7, 8\} \] The complement of \( A \cup B \) with respect to \( U \) is: \[ (A \cup B)' = U - (A \cup B) = \{1, 9\} \]
Step 3: Find \( A' \) and \( B' \).
The complement of \( A \) is: \[ A' = U - A = \{1, 3, 5, 7, 9\} \] The complement of \( B \) is: \[ B' = U - B = \{1, 4, 6, 8, 9\} \]
Step 4: Find \( A' \cap B' \).
The intersection of \( A' \) and \( B' \) is: \[ A' \cap B' = \{1, 9\} \]
Step 5: Conclusion.
We have shown that: \[ (A \cup B)' = A' \cap B' = \{1, 9\} \]