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

Answer : Reflexive, symmetric, transitive.

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 : $R=\{(x, y): x \text { and } y \text { work at same place }\}$

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

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

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

Now, let's get back to the actual problem

For Reflexive :

$x$ and $x$  work at same place

Similarly, $y$ and $y$ work at the same place.

So, $R$ is reflexive.

For Symmetric :

It is given that $x$ and $y$ work at the same place.

So, we can say that $y$ and $x$ work at the same place.

So, $R$ is symmetric.

For Transitive:

Let $z$ be a person such that $y$ and $z$ work at the same place.

And we know that $x$ and $y$ work at the same place.

So, we can say that $x$ and $z$ work at the same place.

So, $R$ is Transitive.