Relation and Function Quiz

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Question 1

How many onto (or surjective) functions are there from an n-element (n >= 2) set to a 2-element set?
 

  • 2(2n - 2)
     

  • 2n - 2
     

  • 2n - 1
     

  • 2n
     

Question 2

Consider the binary relation R = {(x, y), (x, z), (z, x), (z, y)} on the set {x, y, z}. Which one of the following is TRUE?

  • R is symmetric but NOT antisymmetric

  • R is NOT symmetric but antisymmetric

  • R is both symmetric and antisymmetric

  • R is neither symmetric nor antisymmetric

Question 3

Let R be the set of all binary relations on the set {1, 2, 3}. Suppose a relation is chosen from R at random. The probability that the chosen relation is reflexive (round off to 3 decimal places) is ________ .

Note - This question was Numerical Type.

  • 0.125

  • 0.25

  • 0.50

  • 0.625

Question 4

Let S denote the set of all functions f: {0,1}4 -> {0,1}. Denote by N the number of functions from S to the set {0,1}. The value of Log2Log2N is ______.

  • 12

  • 13

  • 15

  • 16

Question 5

A relation R is defined on the set of integers as xRy if f(x + y) is even. Which of the following state­ments is true?

  • R is not an equivalence relation

  • R is an equivalence relation having 1 equivalence class

  • R is an equivalence relation having 2 equivalence classes

  • R is an equivalence relation having 3 equivalence classes

Question 6

The binary relation S = ф (empty set) on set A = {1, 2, 3} is :

  • Neither reflexive nor symmetric

  • Symmetric and reflexive

  • Transitive and reflexive

  • Transitive and symmetric

Question 7

Let f : A → B be an injective (one-to-one) function. 

Define g : 2A → 2B as :
g(C) = {f(x) | x ∈ C}, for all subsets C of A.
Define h : 2B → 2A as :
h(D) = {x | x ∈ A, f(x) ∈ D}, for all subsets D of B.

Which of the following statements is always true ?

  • g(h(D)) ⊆ D

  • g(h(D)) ⊇ D

  • g(h(D)) ∩ D = ф

  • g(h(D)) ∩ (B - D) ≠ ф

Question 8

Consider the binary relation: 

S = {(x, y) | y = x+1 and x, y ∈ {0, 1, 2, ...}}

The reflexive transitive closure of S is

  • {(x, y) | y > x and x, y ∈ {0, 1, 2, ... }}

  • {(x, y) | y ≥ x and x, y ∈ {0, 1, 2, ... }}

  • {(x, y) | y < x and x, y ∈ {0, 1, 2, ... }}

  • {(x, y) | y ≤ x and x, y ∈ {0, 1, 2, ... }}

Question 9

Let f: B → C and g: A → B be two functions and let h = f o g. Given that h is an onto function. Which one of the following is TRUE?

  • f and g should both be onto functions.

  • f should be onto but g need not be onto

  • g should be onto but f need not be onto

  • both f and g need not be onto

Question 10

Let R and S be any two equivalence relations on a non-empty set A. Which one of the following statements is TRUE?

  • R ∪ S, R ∩ S are both equivalence relations

  • R ∪ S is an equivalence relation

  • R ∩ S is an equivalence relation

  • Neither R ∪ S nor R ∩ S is an equivalence relation

There are 20 questions to complete.

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