Questions: Reasoning (31005) covers Units 1, 2, 3 Question 6 of 23 Find set A′ ∩ B′ U=1,2,3,4,5,6,7 A=1,3,5,7 B=2,5,7 Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. A′ ∩ B′= (Use a comma to separate answers as needed.) B. A′ ∩ B′ is the empty set.

Solution Steps

To find \( A' \cap B' \), we need to determine the complements of sets \( A \) and \( B \) with respect to the universal set \( U \), and then find their intersection.

1. Find \( A' \): The complement of \( A \) is the set of elements in \( U \) that are not in \( A \).
2. Find \( B' \): The complement of \( B \) is the set of elements in \( U \) that are not in \( B \).
3. Find \( A' \cap B' \): The intersection of the complements \( A' \) and \( B' \) is the set of elements that are in both \( A' \) and \( B' \).

To find the complements of sets \( A \) and \( B \) with respect to the universal set \( U \):

• The universal set is \( U = \{1, 2, 3, 4, 5, 6, 7\} \).
• The set \( A = \{1, 3, 5, 7\} \) has a complement \( A' = U - A = \{2, 4, 6\} \).
• The set \( B = \{2, 5, 7\} \) has a complement \( B' = U - B = \{1, 3, 4, 6\} \).

Next, we find the intersection of the complements \( A' \) and \( B' \):

\[ A' \cap B' = \{2, 4, 6\} \cap \{1, 3, 4, 6\} = \{4, 6\} \]

Final Answer