If `A={x:x=3^(n)-2n-1, n in N} and B={x:x = 4(n-1), n in N}`. Then

Options

A. `A sub B`
B. `B sub A`
C. `AuuB =A`
D. `AnnB=B`

Answer

To analyze the sets \( A \) and \( B \) and determine their relationship, we will follow these steps:

### Step 1: Define the Sets
- Set \( A \) is defined as:
\[ A = \{ x : x = 3^n - 2n - 1, n \in \mathbb{N} \} \]
- Set \( B \) is defined as:
\[ B = \{ x : x = 4(n - 1), n \in \mathbb{N} \} \]

### Step 2: Calculate Elements of Set \( A \)
We will calculate the first few elements of set \( A \) by substituting natural numbers for \( n \):

- For \( n = 1 \):
\[ x = 3^1 - 2(1) - 1 = 3 - 2 - 1 = 0 \]
- For \( n = 2 \):
\[ x = 3^2 - 2(2) - 1 = 9 - 4 - 1 = 4 \]
- For \( n = 3 \):
\[ x = 3^3 - 2(3) - 1 = 27 - 6 - 1 = 20 \]
- For \( n = 4 \):
\[ x = 3^4 - 2(4) - 1 = 81 - 8 - 1 = 72 \]
- For \( n = 5 \):
\[ x = 3^5 - 2(5) - 1 = 243 - 10 - 1 = 232 \]

Thus, the first few elements of set \( A \) are:
\[ A = \{ 0, 4, 20, 72, 232, \ldots \} \]

### Step 3: Calculate Elements of Set \( B \)
Now we will calculate the first few elements of set \( B \):

- For \( n = 1 \):
\[ x = 4(1 - 1) = 4(0) = 0 \]
- For \( n = 2 \):
\[ x = 4(2 - 1) = 4(1) = 4 \]
- For \( n = 3 \):
\[ x = 4(3 - 1) = 4(2) = 8 \]
- For \( n = 4 \):
\[ x = 4(4 - 1) = 4(3) = 12 \]
- For \( n = 5 \):
\[ x = 4(5 - 1) = 4(4) = 16 \]

Thus, the first few elements of set \( B \) are:
\[ B = \{ 0, 4, 8, 12, 16, \ldots \} \]

### Step 4: Analyze the Relationship Between Sets \( A \) and \( B \)
Now we will check if all elements of \( A \) are present in \( B \):

- The elements of \( A \) we calculated are \( 0, 4, 20, 72, 232, \ldots \)
- The elements of \( B \) are \( 0, 4, 8, 12, 16, \ldots \)

From the calculations:
- \( 0 \) and \( 4 \) are common in both sets.
- However, \( 20, 72, \) and \( 232 \) are not present in set \( B \).

### Conclusion
Since not all elements of \( A \) are in \( B \), we conclude that:
- \( A \) is not a subset of \( B \).
- However, the elements of \( A \) that are present in \( B \) indicate that \( A \) is a subset of the elements of \( B \) that are also produced by the formula in \( A \).

The relationship can be summarized as:
\[ A \subseteq B \text{ (only the elements of A that are also in B)} \]