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JEE MainMathematicsSets and Relations

Let S = 1, 2, 3, 4, 5 and T = 1, 2, 3 . A relation R is defined on the power set P(S) of S by X R Y if and only if X T = Y T . Then the relation R is

Options

  1. Aan equivalence relation with exactly 8 equivalence classes
  2. Breflexive and symmetric but not transitive
  3. Can equivalence relation with exactly 4 equivalence classes
  4. Dan equivalence relation with exactly 32 equivalence classes

Correct answer

C. an equivalence relation with exactly 4 equivalence classes

Step-by-step solution

For any X P(S) , X T = X T , so X R X . Thus, R is reflexive. If X R Y , then X T = Y T , which implies Y T = X T , so Y R X . Thus, R is symmetric. If X R Y and Y R Z , then X T = Y T and Y T = Z T . This implies X T = Z T , so X R Z . Thus, R is transitive. Therefore, R is an equivalence relation. The equivalence classes of R are determined by the distinct values of X T for X P(S) . Since X S , the set X T must be a subset of S that contains T . The subsets of S containing T = 1, 2, 3 are formed by adding element

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