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

Answer:  $R_{2}$ is reflexive and symmetric but not 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:

$R_{2} \text { on } Z$ defined by $(a, b) \in R_{2} \Leftrightarrow|a-b| \leq 5$

Solution :

Reflexivity:

Let a be an arbitrary element of $R_{2}$

Then, $a \in R_{2}$

On applying the given condition, we get

$|a-a|=0 \leq 5$

So, $R_{2}$ is reflexive

Symmetry :

Let $(a, b) \in R_{2} \quad|a-b| \leq 5$

[Since  $|a-b|=|b-a|$ ]

Then $|b-a| \leq 5$

$(b, a) \in R_{2}$

So, $R_{2}$ is symmetric.

Transitivity :

Let   $(1,3) \in R_{2} \text { and }(3,7) \in R_{2}$

$|1-3| \leq 5 \text { and }|3-7| \leq 5$

But,    $|1-7| \leq 5$

$(1,7) \neq R_{2}$

So, $R_{2}$ is not transitive.