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 of finding the number of ordered pairs in an equivalence relation \( R \) on a finite set \( A \) with \( n \) elements, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Equivalence Relations**:
An equivalence relation on a set \( A \) is a relation that is reflexive, symmetric, and transitive. This means that for every element \( a \in A \):
- Reflexive: \( (a, a) \in R \)
- Symmetric: If \( (a, b) \in R \), then \( (b, a) \in R \)
- Transitive: If \( (a, b) \in R \) and \( (b, c) \in R \), then \( (a, c) \in R \)

2. **Counting Ordered Pairs**:
- Since \( R \) is an equivalence relation, it can be represented in terms of equivalence classes. Each element in set \( A \) belongs to exactly one equivalence class.
- Let’s denote the number of equivalence classes as \( k \). Each equivalence class can contain one or more elements from \( A \).

3. **Minimum Ordered Pairs**:
- The minimum number of ordered pairs in \( R \) corresponds to the scenario where each element is only related to itself. This gives us \( n \) ordered pairs, which are \( (a_1, a_1), (a_2, a_2), \ldots, (a_n, a_n) \).
- Therefore, the minimum number of ordered pairs in \( R \) is \( n \).

4. **Maximum Ordered Pairs**:
- In the case where every element is related to every other element (i.e., all elements are in a single equivalence class), the number of ordered pairs would be \( n^2 \). This is because for each of the \( n \) elements, there are \( n \) possible pairs.

5. **Conclusion**:
- Hence, the number of ordered pairs in an equivalence relation \( R \) on a finite set \( A \) with \( n \) elements can vary from \( n \) (minimum) to \( n^2 \) (maximum), depending on how the equivalence classes are formed.

### Final Answer:
The number of ordered pairs in \( R \) is at least \( n \) and at most \( n^2 \).