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#### Need solution for RD Sharma maths Class 12 Chapter Relation Exercise 1.1 Question 5 Subquestion (iii) maths textbook solution.

Answer: Transitive neither reflexive nor symmetric.

Hint:

A relation R on set A is

Reflexive relation:

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

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 $\mathrm{a}, \mathrm{b}, \mathrm{c} \in A$

Given :  $a R b \text { if }|a| \leq b$

Solution :

Reflexivity:

Let -a be an arbitrary element of R

Then $-a \in R$

$\Rightarrow|-a| \neq-a$

So, $R$ is not reflexive

Symmetry:

Let  $(a, b) \in R$

\begin{aligned} &|a| \leq b \\ &|b| \lessgtr a \text { for all } a, b \in R \\ &(b, a) \notin R \end{aligned}

So, $R$ is not symmetric.

Transitivity:

\begin{aligned} &\text { Let }(a, b) \in R_{\text {and }}(b, c) \in R \\ &|a| \leq b \text { and }|b| \leq c \text { for } a, b, c \in R \end{aligned}

Multiplying the corresponding sides, we get

\begin{aligned} &|a| \times|b| \leq b c \\ &|a| \leq c \\ &(a, c) \in R \end{aligned}

Thus, $R$ is transitive