To solve the problem, we need to find the number of elements in the complement of the union of sets $A$ and $B$, denoted as $(A \cup B)^c$.

Step-by-step approach:

1. Identify the Universal Set $S$: $S = \{1, 2, 3, \dots, 30\}$ The total number of elements in $S$ is 30.

2. Identify the sets $A$ and $B$:

   • Set $A = \{7, 10, 11, 12, 14, 15, 18, 21, 22, 23, 25, 28, 30\}$
   • Set $B = \{6, 8, 9, 11, 14, 18, 21, 22, 23, 24, 28\}$

3. Union of sets $A$ and $B$ $(A \cup B)$: The union consists of all unique elements in $A$ and $B$. $A \cup B = \{6, 7, 8, 9, 10, 11, 12, 14, 15, 18, 21, 22, 23, 24, 25, 28, 30\}$ There are 17 elements in $A \cup B$.

4. Complement of the union $(A \cup B)^c$: The complement of $A \cup B$ consists of all elements of $S$ that are not in $A \cup B$. The total number of elements in $S$ is 30, and the number of elements in $A \cup B$ is 17. Thus, the number of elements in the complement is: $\text{Number of elements in } (A \cup B)^c = 30 - 17 = 13$

Thus, the number of elements in $(A \cup B)^c$ is 13.

Final Answer:

$n[(A \cup B)^c] = 13$

Would you like further details or have any questions? Here are some related questions to explore:

1. How do you calculate the intersection of two sets?
2. What is the difference between union and intersection in set theory?
3. Can you find the number of elements in the union of two sets with overlapping elements?
4. What does it mean to take the complement of a set in terms of the universal set?
5. How can Venn diagrams help visualize set operations?

Tip: When working with sets, drawing a Venn diagram can help you visualize the relationships between sets like union, intersection, and complement.