KVPY Sample Paper KVPY Stream-SX Model Paper-25

  • question_answer
    Let R be the relation in the set Z of all integers defined by R = {(x, y): x - y is an integer}. Then R is

    A) Reflexive

    B) Symmetric

    C) Transitive

    D) An equivalence relation

    Correct Answer: D

    Solution :

    Here, R = {(x, y): x - y is an integer} is a relation in the set of integers. For reflexivity, put y- x, x - x = 0 which is an integer for all\[x\in Z.\]. So, R is reflexive in Z. For symmetry, let\[(x,y)\in \operatorname{R}.\], then (x -y) is an integer \[\lambda \] (say) and also \[y-x=-\lambda .\]\[(\because \lambda \in \operatorname{Z}\Rightarrow -\lambda \in \operatorname{Z})\]
    \[\therefore \]\[y-x\]is an integer \[\Rightarrow (y,x)\in R\]. So, R is symmetric. For transitivity, let \[(x,y)\in R,\]and \[(y,z)\in R\]  So x-y = integer and y -z = integers, then x-z is also an integer. 
    \[\therefore \]\[(x,z)\in R.\]So R is transitive.

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