Question

Let R be any relation in the set A of human beings in a town at a particular time. If R={(x,y):x and y work at the same place}, then
(a) Reflexive only
(b) Transitive only
(c) An equivalence relation
(d) Symmetric only

Answer 1

Hint:Think of the basic definition of the types of relations given in the question and try to check whether the relation mentioned in the questions satisfies the condition for any type of relation or not.

Complete step-by-step answer:
Before starting with the solution, let us discuss different types of relations. There are a total of 8 types of relations that we study, out of which the major ones are reflexive, symmetric, transitive, and equivalence relation.
Reflexive relations are those in which each and every element is mapped to itself, i.e., $\left( a,a \right)\in R$ . Symmetric relations are those for which, if \[\left( a,b \right)\in R\text{ }\] then $\left( b,a \right)$ must also belong to R. This can be represented as $aRb\Rightarrow bRa$ . Now, transitive relations are those for which, if $\left( a,b \right)\text{ and }\left( b,c \right)\in R$ then $\left( a,c \right)$ must also belong to R, i.e., $\left( a,b \right)\text{ and }\left( b,c \right)\in R\Rightarrow \left( a,c \right)\in R$ .
Now, if there exists a relation, which is reflexive, symmetric, and transitive at the same time, then the relation is said to be an equivalence relation. For example: let us consider a set A=(1,2). Then the relation {(1,2),(2,1),(1,1),(2,2)} is an equivalence relation.
Now let us start with the solution to the above question. The given relation is reflexive as the person will definitely work in the same place as himself. Also,consider person A works with person B, implies that person B works with person A as well, so the relation is symmetric as well. The relation is transitive as well, because if two persons A and B work in the same place and persons B and C work in the same place then it implies persons A and C work in the same place, then this implies that all three persons work at the same place which shows transitive relation. As we have shown that the relation is symmetric, reflexive, and transitive, we can say that the relation is an equivalence relation. Therefore, the answer to the above question is option (c).

Note: Students should note here that for a relation to be a particular type of relation, i.e. reflexive, symmetric or transitive, all its elements must satisfy the required conditions. If we are able to find even a single counter example, i.e. if any of the elements doesn’t satisfy the criteria, then we can’t proceed further. Also, most of the questions as above are either solved by using statements based on observation or taking examples, as we did in the above question.