If `X={4^(n)-3n-1:ninN}andy={9(n-1):ninN}`, then `X uu Y` equals

Options

A. X
B. Y
C. N
D. None of these

Answer

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

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

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

We will compute the first few elements of set \( X \) by substituting natural numbers for \( n \):

- 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 \( Y \)

Now, we will compute the first few elements of set \( Y \):

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

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

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

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

- For \( n = 5 \):
\[ Y_5 = 9(5 - 1) = 9(4) = 36 \]

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 \( Y \)

The union of two sets \( A \) and \( B \) is defined as the set of elements that are in \( A \), in \( B \), or in both. Therefore, we combine the elements of \( X \) and \( Y \):

- From \( X \): \( 0, 9, 54, 243, \ldots \)
- From \( Y \): \( 0, 9, 18, 27, 36, \ldots \)

The union \( X \cup Y \) will include all unique elements from both sets. Notably, every element from \( X \) (which are \( 0, 9, 54, 243, \ldots \)) is also present in \( Y \) (as \( 0 \) and \( 9 \)). Additionally, \( Y \) contains elements \( 18, 27, 36, \ldots \) which are not in \( X \).

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

### Final Answer

Therefore, the union \( X \cup Y \) is equal to:
\[ Y \]