Propositional and First Order Logic.

Question 31

Which one of the following is not equivalent to p←→q

• A
• B
• C
• D

Question 32

Consider the following two statements.

S1: If a candidate is known to be corrupt, then he will not be elected.
S2: If a candidate is kind, he will be elected.

Which one of the following statements follows from S1 and S2 as per sound inference rules of logic?

• If a person is known to be corrupt, he is kind
• If a person is not known to be corrupt, he is not kind
• If a person is kind, he is not known to be corrupt
• If a person is not kind, he is not known to be corrupt

Question 33

Which one of the following well formed formulae is a tautology? 

• ∀x∃yR(x,y)<-->∃y∀xR(x,y)

• (∀x(∃yR(x,y)->S(x,y)])->∀x∃yS(x,y)

• [∀x∃y(P(x,y)->R(x,y)]<->[∀x∃y(~P(x,y)∪R(x,y)]

• ∀x∃yR(x,y)<-->∀y∀xR(x,y)

Question 34

Let # be a binary operator defined as X # Y = X′ + Y′ where X and Y are Boolean variables. Consider the following two statements.

S1: (P # Q) # R = P # (Q # R)
S2: Q # R = R # Q 

Which of the following is/are true for the Boolean variables P, Q and R?

• Only S1 is True
• Only S2 is True
• Both S1 and S2 are True
• Neither S1 nor S2 are True

Question 35

Suppose U is the power set of the set S = {1,2,3,4,5,6}. For any T ∈ U, let |T| denote the number of elements in T and T′ denote the complement of T. For any T, R ∈ U, let T\\R be the set of all elements in T which are not in R. Which one of the following is true?

• A
• B
• C
• D

Question 36

In a room there are only two types of people, namely Type 1 and Type 2. Type 1 people always tell the truth and Type 2 people always lie. You give a fair coin to a person in that room, without knowing which type he is from and tell him to toss it and hide the result from you till you ask for it. Upon asking, the person replies the following:

“The result of the toss is head if and only if I am telling the truth.” 

Which of the following options is correct?

• The result is head
• The result is tail
• If the person is of Type 2, then the result is tail
• If the person is of Type 1, then the result is tai

Question 37

Which of the above arguments are valid?

• P and Q only
• P and R only
• P and S only
• P, Q, R and S

Question 38

Let P(x) and Q(x) be arbitrary predicates. Which of the following statements is always TRUE?

• ((∀x(P(x)∨Q(x))))⟹((∀xP(x))∨(∀xQ(x)))
• (∀x(P(x)⟹Q(x)))⟹((∀xP(x))⟹(∀xQ(x)))
• (∀x(P(x))⟹∀x(Q(x)))⟹(∀x(P(x)⟹Q(x)))
• (∀x(P(x))⇔(∀x(Q(x))))⟹(∀x(P(x)⇔Q(x)))

Question 39

Consider the following expressions: (i) false (ii) Q (iii) true (iv) P ∨ Q (v) ¬Q ∨ P The number of expressions given above that are logically implied by P ∧ (P ⇒ Q) is ______________ [This Question was originally a Fill-in-the-blanks Question]

• 2
• 3
• 4
• 5

Question 40

Which one of the following well-formed formulae in predicate calculus is NOT valid?

• A
• B
• C
• D