A and B are two sets having 3 and 4 elements respectively and having 2 elements in common. Find the number of possible relations which can be defined from A to B.

Explanation

To find the number of possible relations that can be defined from set A to set B, we can follow these steps:

### Step 1: Identify the cardinalities of the sets
- Set A has 3 elements.
- Set B has 4 elements.
- The number of elements in common (A ∩ B) is 2, but this does not affect the calculation of relations from A to B directly.

### Step 2: Calculate the cardinality of the Cartesian product A × B
The number of ordered pairs (a, b) where \( a \in A \) and \( b \in B \) is given by the formula:
\[ |A \times B| = |A| \times |B| \]
Here, \( |A| = 3 \) and \( |B| = 4 \):
\[ |A \times B| = 3 \times 4 = 12 \]

### Step 3: Calculate the number of possible relations from A to B
A relation from set A to set B is a subset of the Cartesian product \( A \times B \). The number of subsets of a set with \( n \) elements is given by \( 2^n \). Therefore, the number of possible relations \( R \) from A to B is:
\[ \text{Number of relations} = 2^{|A \times B|} = 2^{12} \]

### Step 4: Final Calculation
Now we can compute \( 2^{12} \):
\[ 2^{12} = 4096 \]

Thus, the total number of possible relations that can be defined from A to B is **4096**.

### Summary of the Steps:
1. Identify the cardinalities of sets A and B.
2. Calculate the cardinality of the Cartesian product \( A \times B \).
3. Calculate the number of possible relations from A to B using \( 2^{|A \times B|} \).
4. Compute the final result.