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JEE Main202523 Jan 2025Evening ShiftMathematicsSets and RelationsActual

Let X= R R . Define a relation R on X as : (a₁, b₁ ) R (a₂, b₂ ) b₁=b₂ Statement I : R is an equivalence relation. Statement II : For some (a, b) X , the set S= (x, y) X:(x, y) R(a, b) represents a line parallel to y=x . In the light of the above statements, choose the correct answer from the options given below :

Options

  1. ABoth Statement I and Statement II are false
  2. BStatement I is true but Statement II is false
  3. CBoth Statement I and Statement II are true
  4. DStatement I is false but Statement II is true

Correct answer

B. Statement I is true but Statement II is false

Step-by-step solution

Reflexive : (a₁, b ) R (a₁, b₁ ) b₁=b₁ True . array rl Symmetric : & (a₁, b₁ ) R (a₂, b₂ ) b₁=b₂ & (a₂, b₂ ) R (a₁, b₁ ) b₂=b₁ array True aligned Transitive : & (a₁, b₁ ) R (a₂, b₂ ) b₁=b₂ & & (a₂, b₂ ) R (a₃, b₃ ) b₂=b₃ & (a₁, b₁ ) R (a₃ . b₃ ) True b₁=b₃ aligned Hence Relation R is an equivence relation Statement-I is true. For statement - II y = b so False

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