Consider the following statements :
`1.` `Nuu(BnnZ)=(NuuB)nnZ` for any subset `B` of `R`, where `N` is the set of positive integers, `Z` is the set of integers, `R` is the set of real numbers.
`2.` Let `A={n in N : 1 ge n ge 24, n "is a multiple of" 3}`. There exists no subsets `B` of `N` such that the number of elements in `A` is equal to the number of elements in `B`.
Which of the above statemetns is/are correct ? (A) `1` only (B) `2` only (C) Both `1` and `2` (D) Neither `1` nor `2`

Options

A. `1` only
B. `2` only
C. Both `1` and `2`
D. Neither `1` nor `2`

Answer

To solve the given question, we will analyze both statements one by one.

### Statement 1:
**N ∪ (B ∩ Z) = (N ∪ B) ∩ Z for any subset B of R.**

1. **Understanding the Sets:**
- Let \( N \) be the set of positive integers.
- Let \( Z \) be the set of integers (which includes both positive and negative integers).
- Let \( R \) be the set of real numbers.
- Let \( B \) be any subset of \( R \).

2. **Left-Hand Side (LHS):**
- We need to compute \( N \cup (B \cap Z) \).
- First, find \( B \cap Z \). Since \( B \) is a subset of \( R \) and \( Z \) is the set of integers, \( B \cap Z \) will give us the integers that are also in \( B \).
- Now, \( N \cup (B \cap Z) \) will include all positive integers and all integers that are in \( B \).

3. **Right-Hand Side (RHS):**
- We need to compute \( (N \cup B) \cap Z \).
- First, find \( N \cup B \). This will include all positive integers and all elements of \( B \).
- Now, taking the intersection with \( Z \) means we are looking for all integers that are either positive integers or in \( B \).

4. **Conclusion for Statement 1:**
- Both sides represent the same set of integers, thus \( N \cup (B \cap Z) = (N \cup B) \cap Z \) holds true.
- Therefore, Statement 1 is **correct**.

### Statement 2:
**Let A = {n in N : 1 ≤ n ≤ 24, n is a multiple of 3}. There exists no subset B of N such that the number of elements in A is equal to the number of elements in B.**

1. **Finding the Set A:**
- The multiples of 3 between 1 and 24 are: 3, 6, 9, 12, 15, 18, 21, 24.
- Therefore, \( A = \{3, 6, 9, 12, 15, 18, 21, 24\} \).
- The number of elements in \( A \) is 8.

2. **Understanding the Set B:**
- \( B \) is any subset of \( N \) (the positive integers).
- Since \( B \) can be any subset of \( N \), it can also have 8 elements.

3. **Conclusion for Statement 2:**
- It is possible to construct a subset \( B \) of \( N \) that has exactly 8 elements (for example, \( B = \{1, 2, 3, 4, 5, 6, 7, 8\} \)).
- Therefore, there **exists** a subset \( B \) of \( N \) such that the number of elements in \( A \) is equal to the number of elements in \( B \).
- Thus, Statement 2 is **incorrect**.

### Final Conclusion:
- Statement 1 is correct.
- Statement 2 is incorrect.

The correct answer is **(A) 1 only**.