Let A be a finite set containing n elements, then the number of relations on A is

Options

A. `2^(n)`
B. `2^(n^(2))`
C. `n^(2)`
D. `n^(n)`

Answer

To find the number of relations on a finite set \( A \) containing \( n \) elements, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Relations**: A relation on a set \( A \) is a subset of the Cartesian product \( A \times A \).

2. **Finding the Size of \( A \times A \)**:
- The Cartesian product \( A \times A \) consists of all ordered pairs \( (a_i, a_j) \) where \( a_i, a_j \in A \).
- Since \( A \) has \( n \) elements, the number of elements in \( A \times A \) is:
\[ |A \times A| = n \times n = n^2 \]

3. **Finding the Number of Subsets**:
- A relation on \( A \) is any subset of \( A \times A \).
- The number of subsets of a set with \( m \) elements is given by \( 2^m \).
- Therefore, the number of subsets of \( A \times A \) (which corresponds to the number of relations on \( A \)) is:
\[ \text{Number of relations} = 2^{|A \times A|} = 2^{n^2} \]

4. **Conclusion**:
- Hence, the number of relations on the set \( A \) is:
\[ \text{Number of relations on } A = 2^{n^2} \]

### Final Answer:
The number of relations on a finite set \( A \) containing \( n \) elements is \( 2^{n^2} \).

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