Let's solve the problem based on the image provided.

Given:

• Universal Set $U$: $\{1, 2, 3, \dots, 18, 19, 20\}$
• Set $A$: $\{2, 3, 4, 5, 7, 12, 13, 14, 18, 20\}$
• Set $B$: $\{1, 5, 11, 13, 15, 19, 20\}$
• Set $C$: $\{1, 3, 5, 6, 10, 15, 19, 20\}$

We are required to determine the following:

1. $n(A)$ - the number of elements in Set $A$.
2. $n(B)$ - the number of elements in Set $B$.
3. $n(A \cap B)$ - the number of elements in the intersection of Sets $A$ and $B$.
4. $n(B \cap C)$ - the number of elements in the intersection of Sets $B$ and $C$.
5. $n(A \cup C)$ - the number of elements in the union of Sets $A$ and $C$.

Solution:

1. $n(A)$: Count the elements in Set $A$.

   • $A = \{2, 3, 4, 5, 7, 12, 13, 14, 18, 20\}$
   • Number of elements = 10

2. $n(B)$: Count the elements in Set $B$.

   • $B = \{1, 5, 11, 13, 15, 19, 20\}$
   • Number of elements = 7

3. $n(A \cap B)$: Find the intersection of Sets $A$ and $B$, then count the elements.

   • $A \cap B = \{5, 13, 20\}$
   • Number of elements = 3

4. $n(B \cap C)$: Find the intersection of Sets $B$ and $C$, then count the elements.

   • $B \cap C = \{1, 5, 15, 19, 20\}$
   • Number of elements = 5

5. $n(A \cup C)$: Find the union of Sets $A$ and $C$, then count the elements.

   • $A \cup C = \{1, 2, 3, 4, 5, 6, 7, 10, 12, 13, 14, 15, 18, 19, 20\}$
   • Number of elements = 15

Final Answers:

1. $n(A) = 10$
2. $n(B) = 7$
3. $n(A \cap B) = 3$
4. $n(B \cap C) = 5$
5. $n(A \cup C) = 15$

Would you like any further details or explanations on these calculations?

Related Questions:

1. What is the complement of set $A$ in the universal set $U$?
2. How would you find the intersection of sets $A$, $B$, and $C$?
3. What is the union of sets $B$ and $C$?
4. How many elements are in the set $A \cap C$?
5. What is the symmetric difference between sets $A$ and $C$?

Tip:

When dealing with multiple sets, Venn diagrams can be particularly helpful to visualize intersections and unions effectively.