JEE MAIN - Mathematics (2023 - 31st January Morning Shift - No. 6)
Let $$\mathrm{R}$$ be a relation on $$\mathrm{N} \times \mathbb{N}$$ defined by $$(a, b) ~\mathrm{R}~(c, d)$$ if and only if $$a d(b-c)=b c(a-d)$$. Then $$\mathrm{R}$$ is
symmetric and transitive but not reflexive
reflexive and symmetric but not transitive
transitive but neither reflexive nor symmetric
symmetric but neither reflexive nor transitive
Explanation
Given, $(a, b) R(c, d) \Rightarrow a d(b-c)=b c(a-d)$
Symmetric :
(c, d) $R(a, b) \Rightarrow \operatorname{cb}(\mathrm{d}-\mathrm{a})=\mathrm{da}(\mathrm{c}-\mathrm{b}) $
$\Rightarrow$ Symmetric.
Reflexive :
(a, b) R (a, b) $\Rightarrow a b(b-a) \neq b a(a-b) $
$\Rightarrow$ Not reflexive.
Transitive :
$(2,3) \mathrm{R}(3,2)$ and $(3,2) \mathrm{R}(5,30)$ but
$((2,3),(5,30)) \notin \mathrm{R} $
$\Rightarrow$ Not transitive.
Symmetric :
(c, d) $R(a, b) \Rightarrow \operatorname{cb}(\mathrm{d}-\mathrm{a})=\mathrm{da}(\mathrm{c}-\mathrm{b}) $
$\Rightarrow$ Symmetric.
Reflexive :
(a, b) R (a, b) $\Rightarrow a b(b-a) \neq b a(a-b) $
$\Rightarrow$ Not reflexive.
Transitive :
$(2,3) \mathrm{R}(3,2)$ and $(3,2) \mathrm{R}(5,30)$ but
$((2,3),(5,30)) \notin \mathrm{R} $
$\Rightarrow$ Not transitive.
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