Let R be an equivalence relation on a finite set A having n elements. Then the number of ordered pairs in R is

Options

A. Less than n
B. Greater than or equal to n
C. Less than or equal to n
D. None of these

Explanation

To solve the problem, we need to determine the number of ordered pairs in an equivalence relation \( R \) on a finite set \( A \) with \( n \) elements.

### Step-by-Step Solution:

1. **Understanding Equivalence Relations**:
An equivalence relation on a set must satisfy three properties: reflexivity, symmetry, and transitivity.

2. **Reflexivity**:
For a relation \( R \) to be reflexive, every element in set \( A \) must relate to itself. This means that for each element \( x_i \) in \( A \), the ordered pair \( (x_i, x_i) \) must be in \( R \). Since there are \( n \) elements in \( A \), there will be \( n \) ordered pairs of the form \( (x_i, x_i) \) for \( i = 1, 2, \ldots, n \).

3. **Counting Minimum Ordered Pairs**:
Thus, the minimum number of ordered pairs in \( R \) due to reflexivity is \( n \).

4. **Symmetry and Transitivity**:
While symmetry and transitivity do not impose a minimum number of pairs, they do allow for additional pairs to be included in \( R \). Therefore, the total number of ordered pairs can be greater than \( n \).

5. **Conclusion**:
Since \( R \) must include at least the \( n \) pairs from reflexivity, we conclude that the number of ordered pairs in \( R \) is at least \( n \). Thus, the answer is that the number of ordered pairs in \( R \) is greater than or equal to \( n \).

### Final Answer:
The number of ordered pairs in \( R \) is \( \geq n \).