Provide Solution For  R.D. Sharma Maths Class 12 Chapter relations  Exercise 1.1 Question 14  sub question 3 Maths Textbook Solution.

Answer: $R=\{(-5,-6),(-6,-5),(-5,-5)\}$

Hint:

A relation R on set A is

Reflexive relation:

If $(a, a) \in R$for every $a \in A$

Symmetric relation:

If $\left ( a,b \right )$ is true then $\left (b,a\right )$  is also true for every $a, b \in A$

Transitive relation:

$\text { If }(a, b) \text { and }(b, c) \in R \text { , then }(a, c) \in R \text { for every } \mathrm{a}, \mathrm{b}, \mathrm{c} \in A$

Given:

We have to give the example of a relation which is symmetric and transitive but not reflexive.

Solution:

The relation having properties of being symmetric and transitive but not reflexive.

Let $A=\{-5,-6\}$

Define a relation $R$ on $A$ as

$R=\{(-5,-6),(-6,-5),(-5,-5)\}$

Relation $R$ is not reflexive as $(-6,-6) \notin R$

Relation $R$ is symmetric as $(-5,-6),(-6,-5) \in R$

It is seen that $(-5,-6),(-6,-5) \in R$

Also $(-5,-5) \in R$

The relation $R$ is transitive.

Hence relation $R$ is symmetric and transitive but not reflexive.