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: Determine the cardinalities of sets A and B.
- Set A has 5 elements.
- Set B has 3 elements.

### Step 2: Calculate the Cartesian product of sets A and B.
- The Cartesian product \( A \times B \) consists of all possible ordered pairs where the first element is from set A and the second element is from set B.
- The number of elements in the Cartesian product \( |A \times B| \) is calculated as:
\[ |A \times B| = |A| \times |B| = 5 \times 3 = 15 \]

### Step 3: Find the number of subsets of the Cartesian product.
- A relation from set A to set B is defined as any 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 subsets of \( A \times B \) (which represents the number of relations from A to B) is:
\[ \text{Number of relations} = 2^{|A \times B|} = 2^{15} \]

### Final Answer:
- The total number of relations from set A to set B is \( 2^{15} \).

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