Question

Let \( R \) be a reflexive relation on the finite set \( A \) having 10 elements and if \( m \) is the number of ordered pairs in \( R \), then

• A. \( m \geq 10 \)
• B. \( m = 100 \)
• C. \( m = 10 \)
• D. \( m \leq 10 \)

Hint:

For reflexive relations on a finite set, the minimum number of ordered pairs is equal to the size of the set (i.e., the number of elements), since each element must relate to itself.

Answer 1

The Correct Option is A

A relation \( R \) on a set \( A \) is called reflexive if every element of the set \( A \) is related to itself. In other words, for every element \( a \in A \), we must have \( (a, a) \in R \). If the set \( A \) has 10 elements, say \( A = \{ a_1, a_2, \dots, a_{10} \} \), then for the relation \( R \) to be reflexive, the ordered pairs \( (a_i, a_i) \) for \( i = 1, 2, \dots, 10 \) must be included in \( R \). So, there are at least 10 ordered pairs in the relation, corresponding to the reflexive property. Now, a relation \( R \) on a set \( A \) can have additional ordered pairs beyond those required by reflexivity. For each pair of distinct elements \( (a_i, a_j) \) where \( i \neq j \), we may or may not include the pair in \( R \). Since there are 10 elements in \( A \), there are \( 10 \times 10 = 100 \) possible ordered pairs in total. Thus, the number of ordered pairs in \( R \), denoted by \( m \), is at least 10 (due to the reflexive pairs), and it can range up to 100 (if all possible pairs are included). Hence, we can conclude that \( m \geq 10 \). Therefore, the correct answer is \( \boxed{(a)} \).