Let A= {x:x is natural number},
B= {x:x is an even natural number},
C = {x:x is an odd natural number},
D= {x:x is a prime number }. Find
(i) `AcapB`
(ii) `AcapC`
(iii) `AcapD`
(iv) `BcapC`
(v) `BcapD`
(vi) `CuuD`.

Answer

To solve the problem step by step, we will analyze each set and perform the required set operations.

### Given Sets:
- **A** = {x : x is a natural number} = {1, 2, 3, 4, 5, ...}
- **B** = {x : x is an even natural number} = {2, 4, 6, 8, ...}
- **C** = {x : x is an odd natural number} = {1, 3, 5, 7, ...}
- **D** = {x : x is a prime number} = {2, 3, 5, 7, 11, ...}

### Required Operations:

#### (i) \( A \cap B \)
- **Explanation**: We need to find the intersection of sets A and B. The intersection includes elements that are common to both sets.
- **Solution**: Since all even natural numbers (set B) are also natural numbers (set A), we have:
\[ A \cap B = B = \{2, 4, 6, 8, ...\} \]

#### (ii) \( A \cap C \)
- **Explanation**: We need to find the intersection of sets A and C. This will include elements that are common to both sets.
- **Solution**: All odd natural numbers (set C) are also natural numbers (set A), so:
\[ A \cap C = C = \{1, 3, 5, 7, ...\} \]

#### (iii) \( A \cap D \)
- **Explanation**: We need to find the intersection of sets A and D. This includes elements that are common to both sets.
- **Solution**: All prime numbers (set D) are also natural numbers (set A), thus:
\[ A \cap D = D = \{2, 3, 5, 7, 11, ...\} \]

#### (iv) \( B \cap C \)
- **Explanation**: We need to find the intersection of sets B and C. This will include elements that are common to both sets.
- **Solution**: Set B contains even natural numbers and set C contains odd natural numbers. There are no common elements:
\[ B \cap C = \emptyset \]

#### (v) \( B \cap D \)
- **Explanation**: We need to find the intersection of sets B and D. This will include elements that are common to both sets.
- **Solution**: The only even prime number is 2, which is in both sets:
\[ B \cap D = \{2\} \]

#### (vi) \( C \cup D \)
- **Explanation**: We need to find the union of sets C and D. This will include all elements from both sets.
- **Solution**: The union will consist of all odd natural numbers and all prime numbers. The distinct elements are:
\[ C \cup D = \{1, 2, 3, 5, 7, 9, 11, 13, ...\} \]

### Final Answers:
1. \( A \cap B = B \)
2. \( A \cap C = C \)
3. \( A \cap D = D \)
4. \( B \cap C = \emptyset \)
5. \( B \cap D = \{2\} \)
6. \( C \cup D = \{1, 2, 3, 5, 7, 9, 11, 13, ...\} \)