To find the number of elements in the union of sets $A$ and $B$, denoted as $n(A \cup B)$, we count all the unique elements within both sets $A$ and $B$ without double-counting those that appear in the intersection.

Elements in $A$ (from the diagram):

• 9, 15, $d$, $b$, 5, $c$

Elements in $B$ (from the diagram):

• $x$, 31, 12, $f$, 5, $c$

Union of $A$ and $B$ (combine all unique elements):

• $A \cup B = \{9, 15, d, b, 5, c, x, 31, 12, f\}$

Now, let's count these unique elements:

• $9, 15, d, b, 5, c, x, 31, 12, f$

This gives us a total of 10 elements.

Thus, $n(A \cup B) = 10$.

Would you like to explore anything else related to this problem?

Further Questions:

1. What is $n(A \cap B)$, the number of elements in the intersection of $A$ and $B$?
2. How would you calculate $n(A)$ and $n(B)$ individually?
3. What is the number of elements in the set difference $A - B$?
4. What are the elements in the symmetric difference $A \triangle B$?
5. How many elements are there in the universal set $U$?

Tip: When working with unions and intersections, always pay attention to overlapping elements to avoid double-counting.