A is a set of all the prime numbers less than 20, B is a set of all the even natural numbers less than 20 and C is a set of all the odd natural numbers. Which of the following is/are true statements ?
(A) `A nn B` is a singleton set.
(B) B and C are disjoint sets.
(C) `A nn C` is an infinite set.

Options

A. Only (B)
B. Both (A) and (B)
C. Both (B) and (C)
D. (A), (B) and (C)

Explanation

To solve the problem, we need to first define the sets A, B, and C based on the given information.

### Step 1: Define the sets
1. **Set A**: Prime numbers less than 20.
- The prime numbers less than 20 are: {2, 3, 5, 7, 11, 13, 17, 19}.
- So, A = {2, 3, 5, 7, 11, 13, 17, 19}.

2. **Set B**: Even natural numbers less than 20.
- The even natural numbers less than 20 are: {2, 4, 6, 8, 10, 12, 14, 16, 18}.
- So, B = {2, 4, 6, 8, 10, 12, 14, 16, 18}.

3. **Set C**: Odd natural numbers.
- The odd natural numbers are: {1, 3, 5, 7, 9, 11, 13, 15, 17, 19, ...} (this set is infinite).
- So, C = {1, 3, 5, 7, 9, 11, 13, 15, 17, 19, ...}.

### Step 2: Evaluate the statements
Now we will evaluate each of the statements given in the question.

#### Statement (A): A ∩ B is a singleton set.
- **Intersection of A and B**: A ∩ B = {2, 3, 5, 7, 11, 13, 17, 19} ∩ {2, 4, 6, 8, 10, 12, 14, 16, 18} = {2}.
- Since the intersection contains only one element (2), A ∩ B is indeed a singleton set.
- **Conclusion**: Statement (A) is true.

#### Statement (B): B and C are disjoint sets.
- **Disjoint sets**: Two sets are disjoint if they have no elements in common.
- **Elements of B**: {2, 4, 6, 8, 10, 12, 14, 16, 18}.
- **Elements of C**: {1, 3, 5, 7, 9, 11, 13, 15, 17, 19, ...}.
- The only even number in B is 2, which is not in C. All other numbers in B are even, while all numbers in C are odd.
- Therefore, B and C have no elements in common.
- **Conclusion**: Statement (B) is true.

#### Statement (C): A ∩ C is an infinite set.
- **Intersection of A and C**: A ∩ C = {2, 3, 5, 7, 11, 13, 17, 19} ∩ {1, 3, 5, 7, 9, 11, 13, 15, 17, 19, ...} = {3, 5, 7, 11, 13, 17, 19}.
- The intersection contains the elements {3, 5, 7, 11, 13, 17, 19}, which is a finite set with 7 elements.
- **Conclusion**: Statement (C) is false.

### Final Conclusion
- Statement (A) is true.
- Statement (B) is true.
- Statement (C) is false.