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

Neither reflexive, nor symmetric, not transitive.

Hints :

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

Given :

$R=\left\{(x, y): x\; {\text {is wife of }} y\right\}$

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; $a R b \Rightarrow b R a \text { for all } a, b \in A$

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

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

For Reflexive:

$x$ is not wife of $x$ and $y$ is not wife of $y$

So, $R$ is not reflexive.

For Symmetric:

$x$ is the wife of $y$ but $y$ is not the wife of $x$.

So, $R$ is not symmetric.

For Transitive:

Let $z$ be a person; $\mathrm{z} \in A$ such that $y$ is the wife of $z$ .

And it is given that $x$ is the wife of $y$ but this case is not possible. Also, here we can’t show x is the wife of z.

So, R is not transitive.