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JEE Main202331 Jan 2023Morning ShiftMathematicsSets and RelationsActual

Let R be a relation on N × N defined by a , b R c , d if and only if a d b - c = b c a - d . Then R is

Options

  1. Asymmetric but neither reflexive nor transitive
  2. Btransitive but neither reflexive nor symmetric
  3. Creflexive and symmetric but not transitive
  4. Dsymmetric and transitive but not reflexive

Correct answer

A. symmetric but neither reflexive nor transitive

Step-by-step solution

Given, a , b R c , d ⇒ a d b - c = b c a - d ⇒ a , b R c , d ⇔ b - c b c = a - d a d ⇒ 1 c - 1 b = 1 d - 1 a ⇒ 1 a - 1 b = 1 d - 1 c Reflexive: a , b R a , b ⇒ 1 a - 1 b = 1 b - 1 a It is false. So, given relation is not reflexive. Symmetric: a , b R c , d ⇒ 1 a - 1 b = 1 d - 1 c ⇒ 1 c - 1 d = 1 b - 1 a ∴ c , d R a , b It is symmetric. Transitive: a , b R c , d ⇒ 1 a - 1 b = 1 d - 1 c c , d R e , f = 1 c - 1 d = 1 e - 1 f 1 a - 1 b = 1 e - 1 f ⇒ a ,

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