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JEE Main202427 Jan 2024Morning ShiftMathematicsSets and RelationsActual

Let S = 1 , 2 , 3 , … , 10 . Suppose M is the set of all the subsets of S , then the relation R = ( A , B ) : A ∩ B ≠ ϕ ; A , B ∈ M is :

Options

  1. Asymmetric and reflexive only
  2. Breflexive only
  3. Csymmetric and transitive only
  4. Dsymmetric only

Correct answer

D. symmetric only

Step-by-step solution

Given: S = 1 , 2 , 3 , … , 10 and R = ( A , B ) : A ∩ B ≠ ϕ ; A , B ∈ M For Reflexive, M is subset of ' S ' ⇒ ϕ ∈ M for ϕ ∩ ϕ = ϕ But it is given that relation is A ∩ B ≠ ϕ . So, it is not reflexive. For symmetric, If R B A ⇒ A ∩ B ≠ ϕ , then R A B ⇒ B ∩ A ≠ ϕ , So, it is symmetric. For transitive, If A = ( 1 , 2 ) , ( 2 , 3 ) , B = ( 2 , 3 ) , ( 3 , 4 ) then C = ( 3 , 4 ) , ( 5 , 6 ) . ⇒ R B A , R C B is applicable but R C A is not followed. So, it not transitive Hence, the given relation is symmetric only.

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