`A and B` are two sets having `3 and 4` elements respectively and having 2 elements in common The number of relations which can be defined from A to B is. (i) `2^5` (ii) `2^10 -1` (iii)`2^12 -1` (iv) none of these

Options

A. `2^(5)`
B. `2^(10)-1`
C. `2^(12)-1`
D. none of these

Answer

To find the number of relations that can be defined from set A to set B, we can use the formula for the number of relations between two sets.

### Step-by-Step Solution:

1. **Identify the number of elements in each set:**
- Let set A have \( m = 3 \) elements.
- Let set B have \( n = 4 \) elements.

2. **Understand the concept of relations:**
- A relation from set A to set B is a subset of the Cartesian product \( A \times B \).
- The number of elements in the Cartesian product \( A \times B \) is given by \( m \times n \).

3. **Calculate the number of elements in the Cartesian product:**
- Since \( m = 3 \) and \( n = 4 \), the number of elements in \( A \times B \) is:
\[ m \times n = 3 \times 4 = 12 \]

4. **Determine the number of possible relations:**
- The number of relations from set A to set B is given by \( 2^{(m \times n)} \).
- Therefore, the number of relations is:
\[ 2^{12} \]

5. **Evaluate the options:**
- The options provided are:
- (i) \( 2^5 \)
- (ii) \( 2^{10} - 1 \)
- (iii) \( 2^{12} - 1 \)
- (iv) none of these
- Since we calculated \( 2^{12} \), we see that none of the options match \( 2^{12} \).

6. **Conclusion:**
- Therefore, the correct answer is option (iv) none of these.