Use the given information to fill in the number of elements for each region in the Venn diagram. n(A U B) = 33, n(A n B) = 8, n(A) = 24, and n(A' U B) = 40 The number of elements in region x is the number of elements in region y is , the number of el the number of elements in region z is (Type whole numbers.) and the number of elements in region w is

To solve this problem, let's analyze the Venn diagram step-by-step using the given information:

1. **Given Information:**
   - \( n(A \cup B) = 33 \): This is the number of elements in the union of sets \(A\) and \(B\).
   - \( n(A \cap B) = 8 \): This is the number of elements in the intersection of sets \(A\) and \(B\).
   - \( n(A) = 24 \): This is the total number of elements in set \(A\).
   - \( n(A' \cup B) = 40 \): This is the number of elements in the union of the complement of set \(A\) and set \(B\).

2. **Determine Elements in Each Region:**

   Let's denote the regions as follows for the Venn diagram with two sets \(A\) and \(B\):

   - **Region X**: Elements in \(A\) only (i.e., \(A - B\)).
   - **Region Y**: Elements in both \(A\) and \(B\) (i.e., \(A \cap B\)).
   - **Region Z**: Elements in \(B\) only (i.e., \(B - A\)).
   - **Region W**: Elements in neither \(A\) nor \(B\) (i.e., \(A' \cap B'\)).

3. **Calculate Elements in Each Region:**

   - **Region Y (Intersection of \(A\) and \(B\))**:
     \[
     n(Y) = n(A \cap B) = 8
     \]
   
   - **Region X (Only in \(A\))**:
     \[
     n(X) = n(A) - n(A \cap B) = 24 - 8 = 16
     \]
   
   - **Region Z (Only in \(B\))**:
     To find \(n(Z)\), we use the formula for the union:
     \[
     n(A \cup B) = n(A) + n(B) - n(A \cap B)
     \]
     Solving for \(n(B)\):
     \[
     33 = 24 + n(B) - 8 \implies n(B) = 17
     \]
     Therefore, \(n(Z)\) is:
     \[
     n(Z) = n(B) - n(A \cap B) = 17 - 8 = 9
     \]

   - **Region W (Neither \(A\) nor \(B\))**:
     Using \(n(A' \cup B)\):
     \[
     n(A' \cup B) = n(U) - n(A \cap B')
     \]
     Let's find \(n(A \cap B')\). We know that \(A'\) is the complement of \(A\), so we look at the number of elements outside of

Learn how to solve a set theory problem involving elements in different regions of a Venn diagram. Understand concepts such as union, intersection, and complements of sets with step-by-step explanations. Suitable for high school students.

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