Sets And Functions Multiple Choice Questions & Answers

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Sets And Functions Multiple Choice Questions & Answers

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Questions and Answers
  • 1. 
    Let f : X → X such that f(f (x )= x for all x∈ X then
    • A. 

      F is one- to- one and onto

    • B. 

      F is one- to- one but not onto

    • C. 

      F is onto but not one-to -one

    • D. 

      F need not be either one- to -one or onto  

  • 2. 
    Let A be a closed subset of RA≠∅  and   A≠R . Then A is
    • A. 

      The closure of the interior of A

    • B. 

      A countable set

    • C. 

      A compact set

    • D. 

      Not open

  • 3. 
    Which of the following is/are true?
    • A. 

        (1+ 1 /n) n +1 → e   as n →∞

    • B. 

      (1+ 1 /n+1 )n →e  as n→∞ 

    • C. 

      (1+ 1 /n )n 2 →e  as n→∞ 

    • D. 

      (1+ 1 /n 2) n →e  as n→∞ 

  • 4. 
    Let X⊂R  be an infinite countable bounded subset  of R  which of the statements is true
    • A. 

      X cannot be compact

    • B. 

      X contains an interior point

    • C. 

      X may be closed

    • D. 

      Closure of X is countable

  • 5. 
    Which is compact in R n ?
    • A. 

      {x1,x2,x3,……….xn   : xi<1,   1≤i≤n}  

    • B. 

      {x 1 , x 2 , x 3 ,………. x n   : x 1 + x 2 + x 3 ……. x n =0}

    • C. 

      {x 1 , x 2 , x 3 ,………. x n : x i ≥0,  1≤i≤n}

    • D. 

      {x 1 , x 2 , x 3 ,………. x n :   1≤x i ≤2,  1≤i≤n }

  • 6. 
    Let I={1}∪{2} for x∈R let ϕ (x) =dist {x,I} =Inf{ |x-y |:y∈I} then is
    • A. 

      Discontinuous somewhere

    • B. 

      Continuous on R but differentiable only at x=1

    • C. 

      Continuous on R but differentiable only at x=1,2

    • D. 

      Continuous on R but not differentiable only at   x=1, 3/ 2 ,2

  • 7. 
     Suppose f : R→R  is a function that satisfies  |f(x) -f(y)| ≤ |x-y| β, β>0 then
    • A. 

      If β=1 then f is differentiable

    • B. 

      If β>0 then f is uniform continuous

    • C. 

      If β>1  then f is constant function

    • D. 

      F must be a polynomial

  • 8. 
    Which of the following subsets of  R2  is /are convex
    • A. 

      {(x,y):  |x|≤5 , |y|≤10}

    • B. 

      {(x,y) :   x 2 + y 2 =1} 

    • C. 

      {(x,y) :  y ≥ x 2 }

    • D. 

      {(x,y) :  y≤ x 2 }

  • 9. 
    Consider the set X={(-∞,0)∪ 1/n, n ∈ N}⊂R  with the subspace topology. Then
    • A. 

      0 is an isolated point.

    • B. 

      (–2, 0] is an open set

    • C. 

      0 is a limit point of the subset {1 /n ,n∈N}

    • D. 

      (–2, 0) is an open set

  • 10. 
    Let G 1 and G 2 be two subsets of   R 2 and  f: R 2 →R 2 be a function, then
    • A. 

      F -1 (G 1 ∪ G 2 )= f -1 ( G 1 )∪ f -1 ( G 2 )

    • B. 

      F -1 ( G 1) c = (f -1 ( G 1 )) c

    • C. 

      F (G 1 ∩ G 2) =f (G 1 ) ∩ f ( G 2 )

    • D. 

      If G1 is open and G2 is closed then G1+ G2 = {x+y : x∈ G1,y∈ G2 is neither open nor closed

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