Propositional and First Order Logic.

Question 51

Consider these statements : S1 : ∀ can be distributed over conjunction. S2 : ∃ cannot be distributed over disjunction. S3 : ∀ can be distributed over disjunction. S4 : ∃ can be distributed over conjunction. Which of the following is/are false statement(s) about quantifiers ?

• S2 and S3
• S1 and S4
• S1 , S2 and S4
• S2, S3 and S4.

Question 52

Consider the first-order logic sentence φ ≡ ∃s∃t∃u∀v∀w∀x∀y ψ(s, t, u, v, w, x, y) where ψ(s, t, u, v, w, x, y) is a quantifier-free first-order logic formula using only predicate symbols, and possibly equality, but no function symbols. Suppose φ has a model with a universe containing 7 elements. Which one of the following statements is necessarily true?

• There exists at least one model of φ with universe of size less than or equal to 3
• There exists no model of φ with universe of size less than or equal to 3
• There exists no model of φ with universe size of greater than 7
• Every model of φ has a universe of size equal to 7

Question 53

Let P and Q be two propositions, ¬ (P ↔ Q) is equivalent to: (I) P ↔ ¬ Q (II) ¬ P ↔ Q (III) ¬ P ↔ ¬ Q (IV) Q → P

• Only (I) and (II)
• Only (II) and (III)
• Only (III) and (IV)
• None of the above

Question 54

If T(x) denotes x is a trigonometric function, P(x) denotes x is a periodic function and C(x) denotes x is a continuous function then the statement “It is not the case that some trigonometric functions are not periodic” can be logically represented as

• ¬∃(x) [ T(x) ⋀ ¬P(x) ]
• ¬∃(x) [ T(x) ⋁ ¬P(x) ]
• ¬∃(x) [ ¬T(x) ⋀ ¬P(x) ]
• ¬∃(x) [ T(x) ⋀ P(x) ]

Question 55

The proposition (P⇒Q)⋀(Q⇒P) is a

• tautology
• contradiction
• contingency
• absurdity

Question 56

• (1)

• (2)

• (3)

• (4)

Question 57

Let P, Q, R and S be Propositions. Assume that the equivalences P ⇔ (Q ∨ ¬ Q) and Q ⇔ R hold. Then the truth value of the formula (P ∧ Q) ⇒ ((P ∧ R) ∨ S) is always:

• True
• False
• Same as truth table of Q
• Same as truth table of S

Question 58

“If X, then Y unless Z” is represented by which of the following formulae in propositional logic?

• (X ∧ Y) → ¬ Z
• (X ∧ ¬ Z) → Y
• X → (Y ∧ ¬ Z)
• Y → (X ∧ ¬ Z)

Question 59

Consider the following two well-formed formulas in prepositional logic.

F1 : P ⇒ ¬ P
F2 : (P ⇒ ¬ P) ∨ (¬ P ⇒ P)

Which of the following statements is correct?

• F1 is Satisfiable, F2 is valid
• F1 is unsatisfiable, F2 is Satisfiable
• F1 is unsatisfiable, F2 is valid
• F1 and F2 both are Satisfiable

Question 60

Which one of the following Boolean expressions is NOT a tautology?

• ((a → b) ∧ (b → c)) → (a → c)
• (a ↔ c) →( ¬b → (a ∧ c))
• (a ∧ b ∧ c) → (c ∨ a)
• a → (b → a)