Question

Let R be any relation in the set A of human beings in a town at a particular time. If R=\left\\{ \left( x,y \right):x\,and\,y\,live\,in\,the\,same\,locality \right\\}, Then R is
(a) Not reflexive
(b) Not transitive
(c) Not symmetric
(d) An equivalence relation

Answer 1

Hint: In this equation we will check the condition of reflexive, symmetric and transitive for a relation separately and find conclusions.

Complete step-by-step solution -
In the given question, we have a relation, R=\left\\{ \left( x,y \right):x\,and\,y\,live\,in\,the\,same\,locality \right\\}. In any relation, if for elements in the set, elements are related to itself, then the relation is reflexive. That is, if $$$$ belongs to relation R for all x in set A, then R is a reflexive relation. Now, in a given relation, $\left( x,x \right)$ belongs to R means x and x lives in the same locality. Since both x are the same element, therefore the given condition is true for elements of A.
Therefore, R is reflexive.
Also, in any relation, if x is related to y, such that y is also related to x, then the relation is symmetric. That is, if $\left( x,y \right)$ belongs to R such that $\left( y,x \right)$ also belongs to R, then R is a symmetric relation.
$\left( x,y \right)$ belongs to R means that x and y live in the same locality, then y and x also live in the same locality, that is $\left( y,x \right)$ belongs to R.
Also, in any relation, if x is related to y and y is related to z, such that then x is related to z, then R is a transitive relation. That is, if $\left( x,y \right)$ and $\left( y,z \right)$ belongs to R, such that$\left( x,z \right)$ also belong to R, then R is transitive. Now, in given relation,
$\left( x,y \right)$ and $\left( y,z \right)$ belong to R means x and y lives in the same locality and y and z also belong to the same locality. That is x, y and z live in the same locality. Then, we can say x and y live in the same locality, that is $\left( x,z \right)$ belongs to R.
Therefore, R is transitive.
Hence, R is an equivalence relation, that is, reflexive, symmetric and transitive. So, the correct option is (d).

Note: In this type of question, when writing tabular form of relation is not possible, we consider examples of situations to solve the question. We need to remember the definitions of reflexive, symmetric and transitive relation.