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JEE Main & Advanced AIEEE Solved Paper-2008

  • question_answer
    Let R be the real line. Consider the following subsets of the plane \[R\times R\]:D5)                 \[S=\left\{ \left( x,y \right):y=x+1\,\,and\,\,0<x<2 \right\}\]                 \[T=\left\{ \left( x,y \right):x-y\,is\,an\,\operatorname{int}eget \right\}\] Which one of the following is true?

    A) S is an equivalence relation on R but T is not

    B) T is an equivalence relation on R but S is not

    C) Neither S nor T is an equivalence relation on R

    D) Both S and T are equivalence relations on R

    Correct Answer: B

    Solution :

                    For S, \[y=x+1\]                 for reflexive \[x=x+1\,\,\,\Rightarrow \,\,\,0=1\] \[\Rightarrow \] S is not reflexive. So S can not be equivalence. For T, \[x-y\in I\], then \[x-x=0\in I\Rightarrow \]T is reflexive. \[x-y\in I\], then \[y-x\in I\Rightarrow \]T is symmetric also. Now \[x-y\in I\] and \[\Rightarrow x-z\in I\Rightarrow \] T is transitive also. Hence T is an equivalence relation.


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