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JEE Main2008MathematicsSets and RelationsActual

Let R be the real line. Consider the following subsets of the plane R R . S= (x, y): y=x+1 and 0 < x < 2 , T= (x, y): x-y is an integer . Which one of the following is true?

Options

  1. Aneither S nor T is an equivalence relation on R
  2. Bboth S and T are equivalence relations on R
  3. CS is an equivalence relation on R but T is not
  4. DT is an equivalence relation on R but S is not

Correct answer

D. T is an equivalence relation on R but S is not

Step-by-step solution

T= (x, y): x-y l as 0 I T is a reflexive relation. If x-y 1 y-x I T is symmetrical also If x-y=l₁ and y-z=l₂ Then x-z=(x-y)+(y-z)=l₁+I₂ I T is also transitive. Hence T is an equivalence relation. Clearly x x+1 (x, x) S S is not reflexive.

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