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  • Provide solution for RD Sharma maths Class 12 Chapter Relations Exercise 1.1 Question 3 Subquestion (i) maths textbook solution.

Provide solution for RD Sharma maths Class 12 Chapter Relations Exercise 1.1 Question 3 Subquestion (i) maths textbook solution.

Answers (1)

Answer: R_{1} is symmetric but neither reflexive nor transitive.

Hint : 

A relation R on set A is

Reflexive relation:

If  (a, a) \in R for every a \in A

Symmetric relation:

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

Transitive relation:

If  (a,b) and  (b, c) \in R, then (a, c) \in R  for every a, b, c \in A

Given :

R_{1} \text { on } \mathrm{Q}_{0} defined by (a, b) \in R_{1} \Leftrightarrow a=\frac{1}{b}

Solution :

Reflexivity:

Let  a be an arbitrary element of R_{1}

Then, a \in R_{1}

a \neq \frac{1}{a} \text { for all } a \in Q_{0}

So, R_{1} is not reflexive

Symmetry:

Let (a, b) \in R_{1}

Therefore, we can write 'a' as a=\frac{1}{b}

b=\frac{1}{a}

Then (b, a) \in R_{1}

So, R_{1} is symmetric.

For Transitive:

Let   (a, b) \in R_{1} \text { and }(b, c) \in R_{1}

\begin{aligned} &a=\frac{1}{b} \text { and } b=\frac{1}{c} \\ &a=\frac{1}{\left(\frac{1}{c}\right)} \Rightarrow c \\ &a \neq \frac{1}{c} \\ &(a, c) \notin R_{1} \end{aligned}

So, R_{1} is not transitive.

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