Set Theory & Algebra

Question 31

The following is the incomplete operation table a 4-element group. 

 *
 e
 a
 b
 c


 e
 e
 a
 b
 c


 a
 a
 b
 c
 e


 b






 c

The last row of the table is

• c a e b 
  
  
  

• c b a e
  

• c b e a
  

• c e a b
  

Question 32

The inclusion of which of the following sets into 

S = {{1, 2}, {1, 2, 3}, {1, 3, 5}, (1, 2, 4), (1, 2, 3, 4, 5}}

is necessary and sufficient to make S a complete lattice under the partial order defined by set containment ?

• {1}, {1, 3}
  

• {1}, {2, 3}
  

• {1}
  

• {1}, {1, 3}, (1, 2, 3, 4}, {1, 2, 3, 5)
  

Question 33

How many graphs on n labeled vertices exist which have at least (n² - 3n)/2 edges ?

• A
• B
• C
• D

Question 34

Consider the set ∑* of all strings over the alphabet ∑ = {0, 1}. ∑* with the concatenation operator for strings

• does not form a group
• forms a non-commutative group
• does not have a right identity element
• forms a group if the empty string is removed from ∑*

Question 35

Let (5, ≤) be a partial order with two minimal elements a and b, and a maximum element c.

Let P : S → {True, False} be a predicate defined on S.
Suppose that P(a) = True, P(b) = False and 
P(x) ⇒ P(y) for all x, y ∈ S satisfying x ≤ y, 
where ⇒ stands for logical implication.

Which of the following statements CANNOT be true ?

• P(x) = True for all x ∈ S such that x ≠ b
• P(x) = False for all x ∈ S such that x ≠ a and x ≠ c
• P(x) = False for all x ∈ S such that b ≤ x and x ≠ c
• P(x) = False for all x ∈ S such that a ≤ x and b ≤ x

Question 36

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 37

Let ∑ = (a, b, c, d, e) be an alphabet. We define an encoding scheme as follows : g(a) = 3, g(b) = 5, g(c) = 7, g(d) = 9, g(e) = 11. Which of the following numbers is the encoding h of a non-empty sequence of strings ?

• 2⁷ 3⁷ 5⁷
• 2⁸ 3⁸ 5⁸
• 2⁹ 3⁹ 5⁹
• 2¹⁰ 5¹⁰ 7¹⁰

Question 38

Which of the following is true?

• The set of all rational negative numbers forms a group under multiplication.
• The set of all non-singular matrices forms a group under multiplication.
• The set of all matrices forms a group under multiplication.
• Both (2) and (3) are true.

Question 39

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 40

Consider the following relations:

R1(a,b) if (a+b) is even over the set of integers
R2(a,b) if (a+b) is odd over the set of integers
R3(a,b) if a.b > 0 over the set of non-zero rational numbers
R4(a,b) if |a - b| <= 2 over the set of natural numbers

Which of the following statements is correct?

• R1 and R2 are equivalence relations, R3 and R4 are not

• R1 and R3 are equivalence relations, R2 and R4 are not

• R1 and R4 are equivalence relations, R2 and R3 are not

• R1, R2, R3 and R4 are all equivalence relations