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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.

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


 A relation R on set A is

 Reflexive relation:

If (a, a) \in Rfor 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


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


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.



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