Questions: Use the given information to fill in the number of elements for each region in the Venn diagram. n(A ∪ B)=51, n(A ∩ B)=7, n(A)=34, and n(A′ ∪ B′)=73 The number of elements in region x is , the number of elements in region y is , the number of elements in region z is , and the number of elements in region w is (Type whole numbers.)

Solution Steps

We are given that $n(A) = 34$ and $n(A \cap B) = 7$. Region x represents the elements that are in A but not in B. So, $x = n(A) - n(A \cap B) = 34 - 7 = 27$.

Region y represents the intersection of A and B, so $y = n(A \cap B) = 7$.

We are given that $n(A \cup B) = 51$. Since $n(A \cup B) = n(A) + n(B) - n(A \cap B)$, we have $51 = 34 + n(B) - 7$. Therefore, $n(B) = 51 - 34 + 7 = 24$. Region z represents the elements that are in B but not in A. So, $z = n(B) - n(A \cap B) = 24 - 7 = 17$.

We are given $n(A' \cup B') = 73$. By De Morgan's law, $A' \cup B' = (A \cap B)'$. So, $n(A' \cup B') = n(U) - n(A \cap B)$. Therefore, $73 = n(U) - 7$, which means $n(U) = 73 + 7 = 80$. Region w represents the elements that are not in A or B. $w = n(U) - n(A \cup B) = 80 - 51 = 29$.

Final Answer