Question

Two points A and B in a plane are related if OA = OB, where O is a fixed point. This relation is;
A. Reflexive but not symmetric
B. Symmetric but not transitive
C. An equivalence relation
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 solution -
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 know different types of relations. We will check the given relation for these.
Now, we have been given a relation between two points A and B that the points are related if OA = OB, where O is a fixed point. This situation can be represented in the diagram as

Now, we will first check for reflexivity if we have ARA if OA = OA which is always true. Hence, the relation is reflexive.
Now, for symmetric we have if ARB that is OA = OB.
Now, we can write it as,
OB = OA
Therefore we have BRA.
Hence, $ARB\Rightarrow BRA$. Therefore, the relation is symmetric.
Now, for transitive relations we have if,
ARB that is OA = OB …..(1)
BRC that is OB = OC …..(2)
Now, from (1) and (2) we have,
OA = OC
Therefore, ARC
Since, $ARB\ \text{and}\ BRC\Rightarrow ARC$. Hence, the relation is transitive also.
Now, we know if a relation is reflexive, symmetric and transitive then the relation is an equivalence. Therefore, the given relation is an equivalence relation.
Hence, the correct option is (C).

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.