Let R be a reflexive relation on a finite set A having n elements and let there be m ordered pairs in R, then

Options

A. `m ge n`
B. `m le n`
C. `m = n`
D. none of these

Answer

To solve the problem, we need to analyze the properties of reflexive relations on a finite set. Let's break it down step by step.

### Step-by-Step Solution:

1. **Understanding Reflexive Relations**:
A relation \( R \) on a set \( A \) is called reflexive if every element in \( A \) is related to itself. This means that for every element \( a \in A \), the ordered pair \( (a, a) \) must be in \( R \).

2. **Counting Ordered Pairs**:
Given that the set \( A \) has \( n \) elements, we can conclude that there are \( n \) ordered pairs of the form \( (a_i, a_i) \) for each element \( a_i \) in \( A \). Therefore, there are at least \( n \) ordered pairs in the relation \( R \).

3. **Total Ordered Pairs in Relation**:
The total number of ordered pairs in a relation on a set with \( n \) elements can vary. While we must have at least \( n \) pairs due to the reflexive property, \( R \) can also include additional pairs. For example, if \( R \) includes pairs like \( (a_i, a_j) \) where \( i \neq j \), the total number of pairs \( m \) can exceed \( n \).

4. **Conclusion**:
Since \( R \) must include at least the \( n \) pairs \( (a_1, a_1), (a_2, a_2), \ldots, (a_n, a_n) \), we can conclude that:
\[ m \geq n \]
This means that the number of ordered pairs \( m \) in the reflexive relation \( R \) is at least equal to the number of elements \( n \) in the set \( A \).

### Final Result:
Thus, the statement that \( m \) (the number of ordered pairs in \( R \)) is greater than or equal to \( n \) (the number of elements in \( A \)) is true.