If `X={8^n-7n-1:n epsilon N}` and `Y={49(n-1):n epsilon N}`, then

Options

A. `X sube Y`
B. `Y subeX`
C. X=Y
D. None of these

Answer

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

1. \( X = \{ 8^n - 7n - 1 : n \in \mathbb{N} \} \)
2. \( Y = \{ 49(n - 1) : n \in \mathbb{N} \} \)

### Step 1: Understanding the Elements of Set \( X \)

Let's start by calculating the first few elements of set \( X \) for \( n = 1, 2, 3, \ldots \):

- For \( n = 1 \):
\[ 8^1 - 7 \cdot 1 - 1 = 8 - 7 - 1 = 0 \]

- For \( n = 2 \):
\[ 8^2 - 7 \cdot 2 - 1 = 64 - 14 - 1 = 49 \]

- For \( n = 3 \):
\[ 8^3 - 7 \cdot 3 - 1 = 512 - 21 - 1 = 490 \]

- For \( n = 4 \):
\[ 8^4 - 7 \cdot 4 - 1 = 4096 - 28 - 1 = 4067 \]

So, the first few elements of \( X \) are \( \{ 0, 49, 490, 4067, \ldots \} \).

### Step 2: Understanding the Elements of Set \( Y \)

Now, let's calculate the first few elements of set \( Y \):

- For \( n = 1 \):
\[ 49(1 - 1) = 49 \cdot 0 = 0 \]

- For \( n = 2 \):
\[ 49(2 - 1) = 49 \cdot 1 = 49 \]

- For \( n = 3 \):
\[ 49(3 - 1) = 49 \cdot 2 = 98 \]

- For \( n = 4 \):
\[ 49(4 - 1) = 49 \cdot 3 = 147 \]

So, the first few elements of \( Y \) are \( \{ 0, 49, 98, 147, \ldots \} \).

### Step 3: Comparing the Sets \( X \) and \( Y \)

Now we compare the elements of both sets:

- The element \( 0 \) is in both \( X \) and \( Y \).
- The element \( 49 \) is in both \( X \) and \( Y \).
- The element \( 490 \) from \( X \) is not in \( Y \) because \( Y \) contains multiples of \( 49 \) (i.e., \( 49, 98, 147, \ldots \)) and does not reach \( 490 \) until \( n = 11 \) (which gives \( 49 \cdot 10 = 490 \)).
- The elements of \( Y \) such as \( 98 \) and \( 147 \) are not in \( X \).

### Conclusion

Since \( X \) contains elements that are not in \( Y \) and \( Y \) contains elements that are not in \( X \), we conclude that:

1. \( X \) is not a subset of \( Y \).
2. \( Y \) is not a subset of \( X \).
3. \( X \) is not equal to \( Y \).

Thus, the correct answer is that none of the options provided (A, B, C) are correct.

### Final Answer

**None of these.**