MATHEMATICS FOR MARKETERS B CMS05B Given that \( P=\{a, b, c, d, e, f\} \) and \( Q=\{2,4, b, d, f\} \), find the number of proper subsets are in the set \( P \cap Q \) ?

To find the number of proper subsets in the set \( P \cap Q \), we first need to determine the intersection of the sets \( P \) and \( Q \). Given: - \( P = \{a, b, c, d, e, f\} \) - \( Q = \{2, 4, b, d, f\} \) The intersection \( P \cap Q \) consists of the elements that are present in both sets \( P \) and \( Q \). Identifying the common elements: - The elements in \( P \) are \( a, b, c, d, e, f \). - The elements in \( Q \) are \( 2, 4, b, d, f \). The common elements are: - \( b \) - \( d \) - \( f \) Thus, the intersection is: \[ P \cap Q = \{b, d, f\} \] Next, we need to find the number of proper subsets of the set \( P \cap Q \). A proper subset of a set is any subset that is not equal to the set itself. The number of subsets of a set with \( n \) elements is given by \( 2^n \). In this case, the set \( P \cap Q \) has 3 elements (\( b, d, f \)), so the total number of subsets is: \[ 2^3 = 8 \] These subsets include: 1. \( \emptyset \) (the empty set) 2. \( \{b\} \) 3. \( \{d\} \) 4. \( \{f\} \) 5. \( \{b, d\} \) 6. \( \{b, f\} \) 7. \( \{d, f\} \) 8. \( \{b, d, f\} \) (the set itself) Since we are looking for proper subsets, we exclude the subset that is equal to the set itself (\( \{b, d, f\} \)). Therefore, the number of proper subsets is: \[ 8 - 1 = 7 \] Thus, the number of proper subsets in the set \( P \cap Q \) is \( \boxed{7} \).