Sets And Functions Multiple Choice Questions & Answers

10 Questions | By Themes | Updated: Mar 22, 2022 | Attempts: 87

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Sets And Functions Multiple Choice Questions & Answers - Quiz

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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 R, A≠∅ 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 )ⁿ²→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.
  
  {x_₁,x_₂,x_₃,……….x_{_n} : x_{_i}<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 R^²is /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 G₁is open and G₂is closed then G₁+ G₂= {x+y : x∈ G₁,y∈ G₂is neither open nor closed