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To solve this problem, we need to find the following:

1. \( n(A) \) - The number of elements in set \( A \).
2. \( n(\overline{A}) \) - The number of elements in the complement of \( A \) (i.e., elements in \( S \) but not in \( A \)).
3. \( n(B) \) - The number of elements in set \( B \).
4. \( n(A \cap B) \) - The number of elements in the intersection of \( A \) and \( B \).
5. \( n(A \cup B) \) - The number of elements in the union of \( A \) and \( B \).

### Given sets:
- \( S = \{1, 2, 3, \ldots, 20\} \)
- \( A = \{1, 5, 11, 12, 14, 15, 16\} \)
- \( B = \{1, 2, 3, 4, 5, 7, 8, 9, 16, 17, 18, 19\} \)

### Step-by-step calculations:

1. **Number of elements in \( A \) (\( n(A) \))**:
   - \( A \) has 7 elements: \{1, 5, 11, 12, 14, 15, 16\}.
   - \( n(A) = 7 \)

2. **Number of elements in the complement of \( A \) (\( n(\overline{A}) \))**:
   - \( \overline{A} \) consists of the elements in \( S \) but not in \( A \).
   - \( \overline{A} = \{2, 3, 4, 6, 7, 8, 9, 10, 13, 17, 18, 19, 20\} \)
   - \( n(\overline{A}) = 13 \)

3. **Number of elements in \( B \) (\( n(B) \))**:
   - \( B \) has 12 elements: \{1, 2, 3, 4, 5, 7, 8, 9, 16, 17, 18, 19\}.
   - \( n(B) = 12 \)

4. **Number of elements in \( A \cap B \) (intersection)**:
   - \( A \cap B = \{1, 5, 16\} \)
   - \( n(A \cap B) = 3 \)

5. **Number of elements in \( A \cup B \) (union)**:
   - \( A \cup B = \{1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 14, 15, 16, 17, 18, 19\} \)
   - \( n(A \cup B) = 16 \)

### Final Answers:
- \( n(A) = 7 \)
- \( n(\overline{A}) = 13 \)
- \( n(B) = 12 \)
- \( n(A \cap B) = 3 \)
- \( n(A \cup B) = 16 \)

Would you like further details on any of these steps, or do you have any other questions?

### Related Questions:
1. What is the cardinality of \( A - B \)?
2. How do you find the symmetric difference between sets \( A \) and \( B \)?
3. What is the set representation of \( A \cup B \) in roster form?
4. How many elements are in \( \overline{B} \)?
5. What is \( n(S) \) for the universal set given?

### Tip:
To find the number of elements in the union of two sets, use the formula: 
\[
n(A \cup B) = n(A) + n(B) - n(A \cap B)
\]

Let S = {1, 2, 3, ..., 18, 19, 20} be the universal set. Let sets A and B be subsets of S, where: Set A = {1, 5, 11, 12, 14, 15, 16}, Set B = {1, 2, 3, 4, 5, 7, 8, 9, 16, 17, 18, 19}. Determine the following: n(A), n(A'), n(B), n(A ∩ B), n(A ∪ B).

Learn how to determine the number of elements in the union, intersection, and complement of subsets A and B within a universal set S = {1, 2, ..., 20}. Explore key concepts in set theory, including complement, union, and intersection operations.

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