Question

Let R be any relation in the set A of all books in a library of a college given by. If \[R=\left\{ \left( x,y \right):x\,and\,y\,have\,same\,number\,of\,pages \right\}\], Then R is
(a) Not Reflexive
(b) Not Symmetric
(c) Not Transitive
(d) An equivalence relation

Answer 1

Hint: Think of the basic definition of the types of relations given in the figure 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 solution -
In the given question, we have set A of all books in a library of college. A relation R is defined on set A, such as, \[R=\left\{ \left( x,y \right):x\,and\,y\,have\,same\,number\,of\,pages \right\}\].
Now in any relation, if all elements of a set are related to itself, that is, for all x belongs to a set, if $\left( x,x \right)$ belongs to a relation, then R is reflexive.
In a given relation, $\left( x,x \right)$ belongs to R means that x and x have the same number of pages. Since x and x are the same book, so they will clearly have the same number of pages.
Therefore, R is reflexive.
Also, in any relation, if x is related to y such that y is also related to x that is, if $\left( x,y \right)$ belongs to a relation such that $\left( y,x \right)$ also belongs to a relation, then the relation is symmetric.
In a given relation, $\left( x,y \right)$ belongs to R means that x and y will have the same number of pages. Then clearly, y and x will also have the same number of pages, that is $\left( y,x \right)$ belongs to a given relation R. Therefore, R is symmetric.
Also, in any relation, if x is related to y and y is related to z, such that then x is related to z, that is $\left( x,y \right)$ and $\left( y,z \right)$ belongs to R, then $\left( x,z \right)$ also belongs to R, then R is transitive.
In given relation, $\left( x,y \right)$ and $\left( y,z \right)$ belongs to R means that x and y same number of pages and y and z also have same number of pages. That is, x, y and z have the same number of pages. Then, we have, x and z have the same number of pages, that is $\left( x,z \right)$ belongs to R.
Therefore, R is transitive.
Hence, R is reflexive, symmetric and transitive, that is, R an equivalence relation.
Therefore, the correct answer is option (d).

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