Let X be any non-empty set containing n elements, then the number of relations on X is

Options

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

Answer

To find the number of relations on a non-empty set \( X \) containing \( n \) elements, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Relations**: A relation on a set \( X \) is defined as a subset of the Cartesian product \( X \times X \). This means we need to consider all possible pairs of elements from the set \( X \).

2. **Finding the Size of \( X \times X \)**: If \( X \) has \( n \) elements, then the Cartesian product \( X \times X \) will have \( n \times n = n^2 \) elements. This is because each element in \( X \) can be paired with every other element in \( X \) (including itself).

3. **Subsets of \( X \times X \)**: A relation can be any subset of \( X \times X \). The number of subsets of a set with \( m \) elements is given by \( 2^m \). Therefore, the number of subsets of \( X \times X \) (which is \( n^2 \) elements) is \( 2^{n^2} \).

4. **Conclusion**: Thus, the total number of relations on the set \( X \) is \( 2^{n^2} \).

### Final Answer:
The number of relations on a non-empty set \( X \) containing \( n \) elements is \( 2^{n^2} \).

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