Question

find set ( a cap b )

( u = {1, 2, 3, 4, 5, 6, 7} )
( a = {2, 3, 5, 6} )
( b = {1, 3, 5} )

select the correct choice below and, if necessary, fill in the answer box to complete your choice.

a. ( a cap b = ) (\boxed{quad}) (use a comma to separate answers as needed.)

b. ( a cap b ) is the empty set.

Explanation:

Step1: Find \( A' \)

The universal set \( U = \{1, 2, 3, 4, 5, 6, 7\} \) and set \( A = \{2, 3, 5, 6\} \). The complement of \( A \), \( A' \), is the set of elements in \( U \) but not in \( A \). So \( A' = \{1, 4, 7\} \).

Step2: Find \( B' \)

Set \( B = \{1, 3, 5\} \). The complement of \( B \), \( B' \), is the set of elements in \( U \) but not in \( B \). So \( B' = \{2, 4, 6, 7\} \).

Step3: Find \( A' \cap B' \)

The intersection of \( A' \) and \( B' \) is the set of elements common to both \( A' \) and \( B' \). From \( A' = \{1, 4, 7\} \) and \( B' = \{2, 4, 6, 7\} \), the common elements are \( 4, 7 \). So \( A' \cap B' = \{4, 7\} \).

Answer:

\(\{4, 7\}\)