If `X ={(4^(n) -3n -1) | n in N] and Y= {9(n-1)| n in N}`, then `X cup Y` equals to

Options

A. A. X
B. B. Y
C. C. N
D. D. A null set

Answer

To solve the problem, we need to find the union of the two sets \( X \) and \( Y \) defined as follows:

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

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

We will compute the first few elements of set \( X \) by substituting natural numbers \( n = 1, 2, 3, \ldots \):

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

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

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

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

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

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

Next, we will compute the first few elements of set \( Y \) by substituting natural numbers \( n = 1, 2, 3, \ldots \):

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

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

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

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

Thus, the first few elements of set \( Y \) are:
\[ Y = \{ 0, 9, 18, 27, 36, \ldots \} \]

### Step 3: Find the union of sets \( X \) and \( Y \)

Now we will find the union of sets \( X \) and \( Y \):
\[ X \cup Y = \{ 0, 9, 54, 243, \ldots \} \cup \{ 0, 9, 18, 27, 36, \ldots \} \]

### Step 4: Analyze the relationship between \( X \) and \( Y \)

From our calculations, we can see that:
- The elements of \( X \) (0, 9, 54, 243, ...) are specific values derived from the formula \( 4^n - 3n - 1 \).
- The elements of \( Y \) are multiples of 9 (0, 9, 18, 27, ...).

Notably, every element in \( X \) (specifically 0, 9, and 54) appears in \( Y \). Therefore, we can conclude that:
\[ X \subseteq Y \]

### Final Conclusion

Since \( X \) is a subset of \( Y \), the union of the two sets is simply:
\[ X \cup Y = Y \]

Thus, the final answer is:
\[ X \cup Y = Y \]