If `X={4^(n)-3n-1 : n in N}` and `Y={9(n-1) : n in N}`, then

Options

A. `X subY`
B. `X=Y`
C. `Y sub X`
D. none of these

Answer

To solve the problem, we need to analyze the two sets \( X \) and \( Y \) given by:

1. \( X = \{ 4^n - 3n - 1 : n \in \mathbb{N} \} \)
2. \( Y = \{ 9(n - 1) : n \in \mathbb{N} \} \)

We will find the elements of both sets for the first few natural numbers and then determine the relationship between the two sets.

### Step 1: Calculate elements of set \( X \)

We will substitute natural numbers \( n = 1, 2, 3, \ldots \) into the expression for \( X \):

- For \( n = 1 \):
\[ X_1 = 4^1 - 3(1) - 1 = 4 - 3 - 1 = 0 \]

- For \( n = 2 \):
\[ X_2 = 4^2 - 3(2) - 1 = 16 - 6 - 1 = 9 \]

- For \( n = 3 \):
\[ X_3 = 4^3 - 3(3) - 1 = 64 - 9 - 1 = 54 \]

- For \( n = 4 \):
\[ X_4 = 4^4 - 3(4) - 1 = 256 - 12 - 1 = 243 \]

Thus, the first few elements of set \( X \) are:
\[ X = \{ 0, 9, 54, 243, \ldots \} \]

### Step 2: Calculate elements of set \( Y \)

Next, we will substitute natural numbers \( n = 1, 2, 3, \ldots \) into the expression for \( Y \):

- For \( n = 1 \):
\[ Y_1 = 9(1 - 1) = 9 \times 0 = 0 \]

- For \( n = 2 \):
\[ Y_2 = 9(2 - 1) = 9 \times 1 = 9 \]

- For \( n = 3 \):
\[ Y_3 = 9(3 - 1) = 9 \times 2 = 18 \]

- For \( n = 4 \):
\[ Y_4 = 9(4 - 1) = 9 \times 3 = 27 \]

Thus, the first few elements of set \( Y \) are:
\[ Y = \{ 0, 9, 18, 27, \ldots \} \]

### Step 3: Determine the relationship between sets \( X \) and \( Y \)

Now we compare the elements of both sets:

- Elements of \( X \): \( 0, 9, 54, 243, \ldots \)
- Elements of \( Y \): \( 0, 9, 18, 27, \ldots \)

From the comparison, we can see that:
- \( 0 \in Y \)
- \( 9 \in Y \)
- \( 54 \notin Y \)
- \( 243 \notin Y \)

### Conclusion

Since all elements of \( X \) that we found (0 and 9) are also in \( Y \), but not all elements of \( Y \) (like 18 and 27) are in \( X \), we conclude that:

\[ X \subset Y \quad \text{(X is a subset of Y)} \]

### Final Answer

The relationship between the sets is:
\[ X \text{ is a subset of } Y \]