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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

JEE · Math · previous-year question

  1. A.symmetric and transitive but not reflexive
  2. B.reflexive and symmetric but not transitive
  3. C.transitive but neither reflexive nor symmetric
  4. D.symmetric but neither reflexive nor transitivecorrect

Answer

D. 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.

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