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.