Questions: Use a Venn Diagram and the given information to determine the number of elements in the indicated region. n(A)=33, n(B)=15, n(A ∪ B)=42, n(B′)=40. Find n(A ∩ B)′ A. 42 B. 13 C. 49 D. 36

Solution Steps

To solve this problem, we need to use the principle of inclusion-exclusion for sets. We are given the number of elements in sets \(A\) and \(B\), as well as their union. The formula for the union of two sets is \(n(A \cup B) = n(A) + n(B) - n(A \cap B)\). We can rearrange this formula to find \(n(A \cap B)\). Once we have \(n(A \cap B)\), we can find \(n(A \cap B)^{\prime}\) by subtracting \(n(A \cap B)\) from the total number of elements in the universal set, which is given by \(n(B^{\prime}) + n(B)\).

Using the principle of inclusion-exclusion, we can find the intersection of sets \(A\) and \(B\) with the formula: \[ n(A \cap B) = n(A) + n(B) - n(A \cup B) \] Substituting the given values: \[ n(A \cap B) = 33 + 15 - 42 = 6 \]

The total number of elements in the universal set can be calculated using the complement of set \(B\): \[ n(U) = n(B') + n(B) = 40 + 15 = 55 \]

To find the complement of the intersection of sets \(A\) and \(B\), we use: \[ n(A \cap B)' = n(U) - n(A \cap B) \] Substituting the values we found: \[ n(A \cap B)' = 55 - 6 = 49 \]

Final Answer