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

Answers (1)

Answer:

Neither reflexive, nor symmetric nor 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 :

$R=\{(x, y): x \text { is father of } \mathrm{y}\}$

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 $(a, b) \in R$ 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 father of $x$ and $y$ is not father of $y$

So, $R$ is not reflexive

For Symmetric:

It is given that $x$ is the father of $y$ .

But we can say that $y$ is not the father of $x$ .

So $R$ is not symmetric.

For Transitive:

Let $z$ be a person; $z \in A$ such that $y$ is father of $z$ and it is given that x is a father of y.

Then $x$ is grandfather of $z$

So, $R$ is not Transitive.

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