#### Please solve RD Sharma maths Class 12 Chapter Relations Exercise 1.1 Question 1 Sub question (ii) maths text book solution.

Hint :

If $R$ is reflexive $\Rightarrow(a, a) \in R \text { for all } a \in A$

If $R$ is symmetric $\Rightarrow(a, b) \in R \Rightarrow(b, a) \in R_{\text {for all }} a, b \in A$

If $R$ is transitive $\Rightarrow(a, b) \in R \text { and }(b, c) \in R \Rightarrow(a, c) \in R \text { for all } a, b, c \in A$

Given :

$\mathrm{R}=\{(x, y): x \text { and } y \text { live in same locality\} }$

Solution :

A relation $R$ on set A is said to be reflexive if every element of A is related to itself.

Thus, $R$ is reflexive  $\Leftrightarrow(a, a) \in R \; {\text {for all }} a \in A$

A relation $R$ on set A is said to be symmetric relation if $(a, b) \in R \Rightarrow(b, a) \in R$ for all $a, b \in A$

I.e. $a R b \Rightarrow b R a$ for all $a, b \in A$

A relation $R$ on set A is said to be transitive relation if  $(a, b) \in R \; {\text {and }}(b, c) \in R \Rightarrow(c, a) \in R$ for all $a, b, c \in A$

i.e $a R b \text { and } b R c \Rightarrow a R c \text { for all } a, b, c \in A$

For Reflexive:

$x$ and $x$ live in the same locality.

Similarly, $y$ and $y$ live in same locality

So, $R$ is reflexive.

For Symmetric:

$x$ and $y$ live in same locality.

So, we can easily say that $y$ and $x$ live in same locality.

So, $R$ is symmetric.

For Transitive:

Let $z$ be a person; $z \in A$ and $z$ and $y$ live in same locality

And it is given that $x$ and $y$ live in same locality

So, we can say that $x$ and $z$ live in the same locality.

So, $R$ is Transitive.

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