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To solve this problem, we will interpret the provided data and use it to fill in the Venn diagram.

The question provides the following information:
- \( n(A') = 31 \): This represents the number of elements outside set \( A \).
- \( n(B) = 30 \): This represents the number of elements in set \( B \).
- \( n(A' \cup B') = 41 \): This represents the number of elements that are neither in set \( A \) nor in set \( B \).
- \( n(A \cap B) = 16 \): This represents the number of elements in both sets \( A \) and \( B \) (the intersection).

### Venn Diagram Setup

We will label the regions of the Venn diagram as follows:
- Region 1: Outside both sets \( A \) and \( B \) (this represents \( A' \cap B' \)).
- Region 2: Inside \( A \) but not in \( B \) (this represents \( A \cap B' \)).
- Region 3: Inside both \( A \) and \( B \) (this represents \( A \cap B \)).
- Region 4: Inside \( B \) but not in \( A \) (this represents \( B \cap A' \)).

### Step-by-Step Calculation

#### Step 1: Find the number of elements in Region 1.
Since \( n(A' \cup B') = 41 \), the number of elements in Region 1 is given directly as 41. This represents the number of elements outside both sets \( A \) and \( B \).

#### Step 2: Find the number of elements in Region 3.
It is given that \( n(A \cap B) = 16 \), so Region 3 contains 16 elements.

#### Step 3: Find the number of elements in set \( B \) but not in \( A \) (Region 4).
The total number of elements in \( B \) is 30. Out of these 30 elements, 16 are also in \( A \) (Region 3). Therefore, the number of elements in Region 4 is:
\[
n(B \cap A') = 30 - 16 = 14.
\]

#### Step 4: Find the number of elements in set \( A \) but not in \( B \) (Region 2).
To find the number of elements in \( A \), we use the fact that the number of elements outside set \( A \) (i.e., \( n(A') \)) is 31. The total number of elements in the universal set is the sum of the elements in all four regions. Therefore:
\[
n(U) = n(A') + n(A \cap B') + n(A \cap B) + n(B \cap A') = 31 + x + 16 + 14.
\]
We also know that \( n(A' \cup B') = 41 \), meaning the total number of elements in Region 1 is 41.

Finally, we solve for \( x \) (the number of elements in Region 2):
\[
n(U) = 41.


Draw a Venn diagram and use the given information to fill in the number of elements for each region. n(A') = 31, n(B) = 30, n(A' U B') = 41, n(A ∩ B) = 16.

Solve a Venn diagram problem involving sets A and B. Use the provided information: n(A') = 31, n(B) = 30, n(A' ∪ B') = 41, and n(A ∩ B) = 16, to find the number of elements in each region. Suitable for students in Grades 7-10.

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