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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
