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JEE MainMathematicsSets and Relations

Let Z be the set of all integers and Z ^* be the set of all non-zero integers. Two relations R₁ and R₂ are defined as follows: R₁ on Z : a R₁ b if and only if ab is a perfect square. R₂ on Z ^* : a R₂ b if and only if ab is a perfect square. Which of the following statements is correct?

Options

  1. AR₂ is an equivalence relation but R₁ is not
  2. BBoth R₁ and R₂ are equivalence relations
  3. CNeither R₁ nor R₂ is an equivalence relation
  4. DR₁ is an equivalence relation but R₂ is not

Correct answer

A. R₂ is an equivalence relation but R₁ is not

Step-by-step solution

For relation R₁ on Z : Reflexivity: a R₁ a a a = a^2 , which is always a perfect square. So, R₁ is reflexive. Symmetry: a R₁ b ab is a perfect square ba is a perfect square b R₁ a . So, R₁ is symmetric. Transitivity: Let a = 2 , b = 0 , and c = 3 . a R₁ b 2 0 = 0 = 0^2 (a perfect square). b R₁ c 0 3 = 0 = 0^2 (a perfect square). However, a R₁ c 2 3 = 6 , which is not a perfect square. Thus, R₁ is not transitive. Hence, R₁ is not an equivalence relation. For relation R₂ on Z ^* (non-zero integers): Reflexivity and s

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