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KVPY2016MathematicsSets and Relations

For each positive real number , let A_ be the set of all natural numbers n such that ( n+1 - ( n ) < Let A_ ^ c be the complement of A _ in the set of all natural numbers. Then -

Options

  1. AA _ 1 2 , ~A _ 1 3 , ~A _ 2 5 are all finite sets
  2. BA_ 1 3 is a finite set but A_ 1 2 , A_ 2 5 are infinite sets
  3. CA_ 1 2 ^ c , A _ 1 3 ^ c , A _ 2 5 ^ c are all finite sets
  4. DA_ 1 3 , A_ 2 5 are finite sets and A_ 1 2 is an infinite set

Correct answer

C. A_ 1 2 ^ c , A _ 1 3 ^ c , A _ 2 5 ^ c are all finite sets

Step-by-step solution

As n | x+1 - x | 0 There exist infinite natural numbers for which | x+1 - x | 0 Hence A_ 1 2 , A_ 1 3 , A_ 2 5 are all infinite sets

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