To solve the problem, we need to analyze the sets \( X \) and \( Y \) defined as follows:
- \( X = \{ 4^n - 3n - 1 : n \in \mathbb{N} \} \)
- \( Y = \{ 9(n-1) : n \in \mathbb{N} \} \)
### Step 1: Determine the elements of set \( X \)
We will calculate the first few elements of set \( X \) by substituting natural numbers for \( n \).
1. For \( n = 1 \):
\[
4^1 - 3 \cdot 1 - 1 = 4 - 3 - 1 = 0
\]
2. For \( n = 2 \):
\[
4^2 - 3 \cdot 2 - 1 = 16 - 6 - 1 = 9
\]
3. For \( n = 3 \):
\[
4^3 - 3 \cdot 3 - 1 = 64 - 9 - 1 = 54
\]
4. For \( n = 4 \):
\[
4^4 - 3 \cdot 4 - 1 = 256 - 12 - 1 = 243
\]
5. For \( n = 5 \):
\[
4^5 - 3 \cdot 5 - 1 = 1024 - 15 - 1 = 1008
\]
Thus, the first few elements of set \( X \) are:
\[
X = \{ 0, 9, 54, 243, 1008, \ldots \}
\]
### Step 2: Determine the elements of set \( Y \)
Now, we will calculate the first few elements of set \( Y \).
1. For \( n = 1 \):
\[
9(1-1) = 9 \cdot 0 = 0
\]
2. For \( n = 2 \):
\[
9(2-1) = 9 \cdot 1 = 9
\]
3. For \( n = 3 \):
\[
9(3-1) = 9 \cdot 2 = 18
\]
4. For \( n = 4 \):
\[
9(4-1) = 9 \cdot 3 = 27
\]
5. For \( n = 5 \):
\[
9(5-1) = 9 \cdot 4 = 36
\]
Thus, the first few elements of set \( Y \) are:
\[
Y = \{ 0, 9, 18, 27, 36, \ldots \}
\]
### Step 3: Analyze the relationship between sets \( X \) and \( Y \)
From our calculations, we can observe the following:
- The elements of set \( X \) are \( 0, 9, 54, 243, 1008, \ldots \).
- The elements of set \( Y \) are \( 0, 9, 18, 27, 36, \ldots \).
Now, we can see that:
- \( 0 \) is in \( Y \).
- \( 9 \) is in \( Y \).
- \( 54 \) is not in \( Y \).
- \( 243 \) is not in \( Y \).
- \( 1008 \) is not in \( Y \).
However, we can see that all elements of \( X \) that are multiples of \( 9 \) are also present in \( Y \).
### Conclusion
Since all elements of \( X \) that we calculated are multiples of \( 9 \) and \( Y \) contains all multiples of \( 9 \) starting from \( 0 \), we can conclude that:
\[
X \subseteq Y
\]
Thus, the correct answer is that \( X \) is a subset of \( Y \).
