Let A and B be two finite sets such that `A cap B` is a singleton. If n(A) = 6, n(B) = 4, then number of subsets of `A Delta B` is

Options

A. 256
B. 128
C. 512
D. 64

Answer

To solve the problem step by step, we need to find the number of subsets of the symmetric difference of sets A and B, denoted as \( A \Delta B \).

### Step 1: Understand the given information
We are given:
- \( n(A) = 6 \) (the number of elements in set A)
- \( n(B) = 4 \) (the number of elements in set B)
- \( A \cap B \) is a singleton, meaning it contains exactly one element.

### Step 2: Calculate the number of elements in \( A \cap B \)
Since \( A \cap B \) is a singleton, we have:
- \( n(A \cap B) = 1 \)

### Step 3: Use the formula for the union of two sets
To find \( n(A \cup B) \), we use the formula:
\[ n(A \cup B) = n(A) + n(B) - n(A \cap B) \]
Substituting the values we have:
\[ n(A \cup B) = 6 + 4 - 1 = 9 \]

### Step 4: Calculate the number of elements in \( A - B \) and \( B - A \)
- The number of elements in \( A - B \) is:
\[ n(A - B) = n(A) - n(A \cap B) = 6 - 1 = 5 \]
- The number of elements in \( B - A \) is:
\[ n(B - A) = n(B) - n(A \cap B) = 4 - 1 = 3 \]

### Step 5: Calculate the number of elements in \( A \Delta B \)
The symmetric difference \( A \Delta B \) is defined as:
\[ A \Delta B = (A - B) \cup (B - A) \]
Thus, the number of elements in \( A \Delta B \) is:
\[ n(A \Delta B) = n(A - B) + n(B - A) = 5 + 3 = 8 \]

### Step 6: Calculate the number of subsets of \( A \Delta B \)
The number of subsets of a set with \( n \) elements is given by \( 2^n \). Therefore, the number of subsets of \( A \Delta B \) is:
\[ \text{Number of subsets} = 2^{n(A \Delta B)} = 2^8 = 256 \]

### Final Answer
The number of subsets of \( A \Delta B \) is **256**.

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