Question

Let R be a reflexive relation of a finite set A having n elements and let there be m ordered pairs in R. Then,
A. $m\ge n$
B. $m\le n$
C. m = n
D. None of these

Answer 1

Hint: We will be using the concepts of functions and relations to solve the problem. We will be using the definitions of reflexive relation, symmetric relations and transitive relations to verify if each relation holds or not and hence deduce the answer.

Complete step-by-step answer:

Now, we have been given a relation and we have to find whether the relation is reflexive, symmetric, transitive or a combination of these.
Now, we know that reflexive relations are those in which every element is mapped to itself i.e. $\left( a,a \right)\in R$ while symmetric relations are those for which if a R b then b R a. Also, holds and transitive are those relations in which if a R b and b R c then a R c must be held.
Now, we have been given that R be a reflexive relation of a finite set A having n elements and there are m ordered pairs in R.
Now, we want for a reflexive relation a R a where, $a\in A$.
Now, we have n value in set A. Therefore, we have at least n a R a ordered pairs in the relation R. Now, there can be more ordered pairs which satisfy the relation but at least n are there in R. So, we have $m\ge n$.
Hence, the correct option is (A).

Note: To solve these types of questions it is important to note that a R b means that a is related to b by a relation R. Also these types of questions are solved easily by giving examples and counterexamples. Also, we have to check the relation for reflexive, symmetric and transitive relation to check it for equivalence relation.