Statement -I:`X={8n+1:n inN},then XnnY=Y, Y={(2n+1)^2:n in N}` because statement -II:x is a subset of y

Options

A. Statement -1 is True, Statement -2 is True , Statement -2 is a correct explanation for Statement-1
B. Statement -1 is True, Statement -2 is True , Statement -2 is NOT a correct explanation for Statement-1
C. Statement -1 is True, Statement -2 is False.
D. Statement-1 is False, Statement -2 is True.

Answer

To solve the problem, we need to analyze the two statements given about the sets \( X \) and \( Y \).

### Step 1: Define the Sets

1. **Set \( X \)**:
The set \( X \) is defined as:
\[ X = \{ 8n + 1 : n \in \mathbb{N} \} \]
Let's calculate the first few elements of \( X \):
- For \( n = 1 \): \( 8(1) + 1 = 9 \)
- For \( n = 2 \): \( 8(2) + 1 = 17 \)
- For \( n = 3 \): \( 8(3) + 1 = 25 \)
- For \( n = 4 \): \( 8(4) + 1 = 33 \)

Thus, the set \( X \) in roster form is:
\[ X = \{ 9, 17, 25, 33, \ldots \} \]

2. **Set \( Y \)**:
The set \( Y \) is defined as:
\[ Y = \{ (2n + 1)^2 : n \in \mathbb{N} \} \]
Let's calculate the first few elements of \( Y \):
- For \( n = 1 \): \( (2(1) + 1)^2 = 3^2 = 9 \)
- For \( n = 2 \): \( (2(2) + 1)^2 = 5^2 = 25 \)
- For \( n = 3 \): \( (2(3) + 1)^2 = 7^2 = 49 \)
- For \( n = 4 \): \( (2(4) + 1)^2 = 9^2 = 81 \)

Thus, the set \( Y \) in roster form is:
\[ Y = \{ 9, 25, 49, 81, \ldots \} \]

### Step 2: Analyze the Intersection

Now we need to find the intersection of sets \( X \) and \( Y \):
\[ X \cap Y = \{ 9, 25, 49, \ldots \} \]
From our calculations, we see that:
- \( 9 \) is in both \( X \) and \( Y \)
- \( 25 \) is in both \( X \) and \( Y \)
- \( 49 \) is in both \( X \) and \( Y \)

Thus, we can conclude:
\[ X \cap Y = Y \]

### Step 3: Verify the Statements

1. **Statement I**: \( X \cap Y = Y \) is **True**.
2. **Statement II**: \( X \) is a subset of \( Y \) is **False** because \( Y \) is not a subset of \( X \); rather, \( Y \) is a subset of \( X \).

### Conclusion

- The first statement is true: \( X \cap Y = Y \).
- The second statement is false: \( X \) is not a subset of \( Y \).

Thus, the final answer is that Statement I is true and Statement II is false.

---