Set Theory & Algebra

Question 41

Consider the following statements:

S1: There exists infinite sets A, B, C such that 
    A ∩ (B ∪ C) is finite.
S2: There exists two irrational numbers x and y such
    that (x+y) is rational.

Which of the following is true about S1 and S2?

• Only S1 is correct
• Only S2 is correct
• Both S1 and S2 are correct
• None of S1 and S2 is correct

Question 42

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 43

Let P(S) denotes the power set of set S. Which of the following is always true?

• a
• b
• c
• d
• None

Question 44

The binary operator ≠ is defined by the following truth table Which one of the following is true about the binary operator ≠?

• Both commutative and associative
• Commutative but not associative
• Not commutative but associative
• Neither commutative nor associative

Question 45

Suppose L = {p, q, r, s, t} is a lattice represented by the following Hasse diagram: For any x, y ∈ L, not necessarily distinct, x ∨ y and x ∧ y are join and meet of x, y respectively. Let L³ = {(x,y,z): x, y, z ∈ L} be the set of all ordered triplets of the elements of L. Let pr be the probability that an element (x,y,z) ∈ L³ chosen equiprobably satisfies x ∨ (y ∧ z) = (x ∨ y) ∧ (x ∨ z). Then

• Pr = 0
• Pr = 1
• 0 < Pr ≤ 1/5
• 1/5 < Pr < 1

Question 46

Let R be the relation on the set of positive integers such that aRb if and only if a and b are distinct and have a common divisor other than 1. Which one of the following statements about R is True?

• R is symmetric and reflexive but not transitive
• R is reflexive but not symmetric and not transitive
• R is transitive but not reflexive and not symmetric
• R is symmetric but not reflexive and not transitive

Question 47

The cardinality of the power set of {0, 1, 2 . . ., 10} is _________.

• 1024
• 1023
• 2048
• 2043

Question 48

Consider two relations R1(A, B) with the tuples (1, 5), (3, 7) and R1(A, C) = (1, 7), (4, 9). Assume that R(A,B,C) is the full natural outer join of R1 and R2. Consider the following tuples of the form (A,B,C)

  a = (1, 5, null),
  b = (1, null, 7), 
  c = (3, null, 9), 
  d = (4, 7, null), 
  e = (1, 5, 7), 
  f = (3, 7, null), 
  g = (4, null, 9). 

Which one of the following statements is correct?

• R contains a, b, e, f, g but not c, d
• R contains a, b, c, d, e, f, g
• R contains e, f, g but not a, b
• R contains e but not f, g

Question 49

The number of onto functions (surjective functions) from set X = {1, 2, 3, 4} to set Y = {a, b, c} is ________________

• 36

• 64

• 81

• 72

Question 50

Let X and Y denote the sets containing 2 and 20 distinct objects respectively and F denote the set of all possible functions defined from X and Y. Let f be randomly chosen from F. The probability of f being one-to-one is _________

• 0.95
• 0.80
• 0.75
• 0.70