Set A has 5 eleements and set B has 3 elements. Find the number of relations from set A to B.

Answer

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

### Step 1: Understand the definition of a relation
A relation from set A to set B is a subset of the Cartesian product \( A \times B \). This means that each element of set A can be related to any element of set B.

### Step 2: Determine the number of elements in the sets
Given:
- Set A has \( m = 5 \) elements.
- Set B has \( n = 3 \) elements.

### Step 3: Calculate the size of the Cartesian product
The Cartesian product \( A \times B \) will have \( m \times n \) elements. Therefore, the number of elements in \( A \times B \) is:
\[ m \times n = 5 \times 3 = 15 \]

### Step 4: Determine the number of subsets
The number of relations from set A to set B corresponds to the number of subsets of the Cartesian product \( A \times B \). The number of subsets of a set with \( k \) elements is given by \( 2^k \).

### Step 5: Apply the formula
Since we found that \( A \times B \) has 15 elements, the number of relations is:
\[ 2^{15} \]

### Step 6: Calculate the final answer
Now, we can compute \( 2^{15} \):
\[ 2^{15} = 32768 \]

Thus, the number of relations from set A to set B is \( 32768 \).

### Final Answer
The number of relations from set A to set B is \( 32768 \).

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