Let R be a relation on set of all natural numbers given by α b Leftrightarrow α divides b^2 . Whic

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1.Relation and Function

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Let $R$ be a relation on the set of all natural numbers given by $\alpha b \Leftrightarrow \alpha$ divides $b^2$.

Which of the following properties does $R$ satisfy?

$I.$ Reflexivity $II.$ Symmetry $III.$ Transitivity

A

$I$ only

B

$III$ only

C

$I$ and $III$ only

D

$I$ and $II$ only

(KVPY-2017)

Solution

(a)

We have, $a R b: a$ divides $b^2$

For reflexive : $(a, a) \in R$

$\therefore \quad a R a: a$ divides $a^2$.

Hence, $R$ is reflexive.

For symmetric : $(a, b) \in R \Rightarrow(b, a) \in R$

$a$ divides $b^2$ and $b$ not divides $a^2$.

Hence, it is not symmetric.

For transitive : $(a, b) \in R,(b, c) \in R$

$\Rightarrow(a, c) \in R$

$\Rightarrow(8,4): 8$ divides $4^2$

$\Rightarrow(4,2): 4$ divides $2^2$

But $(8 / 2): 8$ not divides $2^2$

$\therefore$ It is not transitive.

Standard 12

Mathematics

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