Set Theory & Algebra

Question 101

Suppose that R₁ and R₂ are reflexive relations on a set A. Which of the following statements is correct ?

• R₁ ∩ R₂ is reflexive and R₁ ∪ R₂ is irreflexive.
• R₁ ∩ R₂ is irreflexive and R₁ ∪ R₂ is reflexive.
• Both R₁ ∩ R₂ and R₁ ∪ R₂ are reflexive.
• Both R₁ ∩ R₂ and R₁ ∪ R₂ are irreflexive.

Question 102

Simplified Boolean equation for the following truth table is:

• F = yz` + y`z
• F = xy` + x`y
• F = x`z + xz`
• F = x`z + xz` + xyz

Question 103

The number of elements in the power set of the set {{A,B},C} is

• 7
• 8
• 3
• 4

Question 104

3. Which of the following is/are not true? (a)The set of negative integers is countable. (b)The set of integers that are multiples of 7 is countable. (c)The set of even integers is countable. (d)The set of real numbers between 0 and ½ is countable.

• (a) and (c)
• (b) and (d)
• (b) only
• (d) only

Question 105

Which of the following property/ies a Group G must hold, in order to be an Abelian group? (a)The distributive property (b)The commutative property (c)The symmetric property Codes:

• (a) and (b)
• (b) and (c)
• (a) and (b)
• (a), (b) and (c)

Question 106

Which one of the following is true?

• R ∩ S = (R ∪ S) − [(R − S) ∪ (S − R)]
• R ∪ S = (R ∩ S) − [(R − S) ∪ (S − R)]
• R ∩ S = (R ∪ S) − [(R − S) ∩ (S − R)]
• R ∩ S = (R ∪ S) ∪ (R − S)

Question 107

The Servlet Response interface enables a servlet to formulate a response for a client using the method __________.

• void log(Exception e, String s)
• void destroy( )
• int get ServerPort( )
• void set ContextType(String type)

Question 108

( G, *) is an abelian group. Then

• x = x ^-1, for any x belonging to G
• x = x², for any x belonging to G
• (x * y )² = x² * y², for any x, y belonging to G
• G is of finite order

Question 109

Which of the relations on {0, 1, 2, 3} is an equivalence relation?

• {(0, 0), (0, 2), (2, 0), (2, 2), (2, 3), (3, 2), (3, 3)}
• {(0, 0) (1, 1) (2, 2) (3, 3)}
• {(0, 0), (0, 1), (0, 2), (1, 0), (1, 1), (1, 2), (2, 0)}
• {(0, 0), (0, 2), (2, 3), (1, 1), (2, 2)}

Question 110

Which of the following is an equivalence relation on the set of all functions from Z to Z?

• {(f, g) | f (x)−g (x) = 1 x ∈ Z}
• {(f, g) | f (0) = g (0) or f (1) = g (1)}
• {(f, g) | f (0) = g (1) and f (1) = g (0)}
• {(f, g) | f (x)−g (x) = k for some k ∈ Z }