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#### Provide Solution For  R. D. Sharma Maths Class 12 Chapter relations  Exercise 1.1 Question 14  sub question 2 Maths Textbook Solution.

Answer:$R=\left\{(a, b): a^{3} \geq b^{3}\right\}$

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 reflexive and transitive but not symmetric.

Solution:

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

Define a relation $R$ in $R$ as:

$R=\left\{(a, b): a^{3} \geq b^{3}\right\}$

Clearly $(a, a) \in R \text { as } a^{3}=a^{3}$

Therefore $R$ is reflexive.

Now $(2,1) \in R\left(\text { as } 2^{3} \geq 1^{3}\right)$

But $(1,2) \notin R\left(\text { as } 1^{3}<2^{3}\right)$

Therefore $R$ is not symmetric.

Now let

\begin{aligned} &(a, b), \quad(b, c) \in R \\ &a^{3} \geq b^{3} \text{and }b^{3} \geq c^{3} \text { then } a^{3} \geq c^{3} \quad(a, c) \in R \end{aligned}

$R$  is transitive.

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