If A = {x : x is a natural number}, B = {x : x is an even natural number}, C = {x : x is an odd natural number} and D = {x : x is a prime number}, Find :
(i) `A cap B`
(ii) `A cap C`
(iii) `A cap D`
(iv) `B cap C`
(v) `B cap D`
(vi) `C cap D`.

Options

A. `AnnB`
B. `AnnC`
C. `BnnD`
D. `CnnD`

Explanation

To solve the problem, we need to find the intersections of the given sets A, B, C, and D. Let's define the sets first:

- \( A = \{ x : x \text{ is a natural number} \} \)
- \( B = \{ x : x \text{ is an even natural number} \} \)
- \( C = \{ x : x \text{ is an odd natural number} \} \)
- \( D = \{ x : x \text{ is a prime number} \} \)

Now, let's find each intersection step by step.

### (i) \( A \cap B \)

**Solution:**
- Set \( A \) contains all natural numbers: \( \{1, 2, 3, 4, 5, \ldots\} \)
- Set \( B \) contains all even natural numbers: \( \{2, 4, 6, 8, \ldots\} \)
- The intersection \( A \cap B \) includes all elements that are in both \( A \) and \( B \), which means all even natural numbers.
- Therefore, \( A \cap B = B = \{2, 4, 6, 8, \ldots\} \).

### (ii) \( A \cap C \)

**Solution:**
- Set \( C \) contains all odd natural numbers: \( \{1, 3, 5, 7, 9, \ldots\} \)
- The intersection \( A \cap C \) includes all elements that are in both \( A \) and \( C \), which means all odd natural numbers.
- Therefore, \( A \cap C = C = \{1, 3, 5, 7, 9, \ldots\} \).

### (iii) \( A \cap D \)

**Solution:**
- Set \( D \) contains all prime numbers: \( \{2, 3, 5, 7, 11, 13, \ldots\} \)
- The intersection \( A \cap D \) includes all elements that are in both \( A \) and \( D \), which means all prime numbers.
- Therefore, \( A \cap D = D = \{2, 3, 5, 7, 11, 13, \ldots\} \).

### (iv) \( B \cap C \)

**Solution:**
- Since \( B \) contains even natural numbers and \( C \) contains odd natural numbers, there are no common elements between these two sets.
- Therefore, \( B \cap C = \emptyset \) (the empty set).

### (v) \( B \cap D \)

**Solution:**
- The only even prime number is \( 2 \), which is in both set \( B \) and set \( D \).
- Therefore, \( B \cap D = \{2\} \).

### (vi) \( C \cap D \)

**Solution:**
- All prime numbers except \( 2 \) are odd, so the intersection of odd natural numbers (set \( C \)) and prime numbers (set \( D \)) will include all odd prime numbers.
- Therefore, \( C \cap D = D - \{2\} = \{3, 5, 7, 11, 13, \ldots\} \).

### Summary of Results:
1. \( A \cap B = \{2, 4, 6, 8, \ldots\} \)
2. \( A \cap C = \{1, 3, 5, 7, 9, \ldots\} \)
3. \( A \cap D = \{2, 3, 5, 7, 11, 13, \ldots\} \)
4. \( B \cap C = \emptyset \)
5. \( B \cap D = \{2\} \)
6. \( C \cap D = \{3, 5, 7, 11, 13, \ldots\} \)