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#### Please Solve R. D. Sharma class 12 Chapter relations Exercise 1.1 Question 8  Maths textbook Solution.

Hence, prove every identity relation on a set is reflexive but the converse is not necessarily true.

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: Every Identity relation is reflexive.

Solution:

Let $A$ be a set $A=\{1,2,3\}$

Then,  $I_{A}=\{(1,1),(2,2),(3,3)\}$

Identity relation is reflexive, since $(a, a) \in A \quad \forall a$

The converse of it need not be necessarily true.

Counter example:

Consider the set $A=\{1,2,3\}$

Here,

Relation$R=\{(1,1),(2,2),(3,3),(2,1),(1,3)\}$ is reflexive on $A$

However, $R$ is not an identity relation.