Propositional and First Order Logic.

Question 21

What is the first order predicate calculus statement equivalent to the following? Every teacher is liked by some student

• ∀(x) [teacher (x) → ∃ (y) [student (y) → likes (y, x)]]
• ∀ (x) [teacher (x) → ∃ (y) [student (y) ^ likes (y, x)]]
• ∃ (y) ∀ (x) [teacher (x) → [student (y) ^ likes (y, x)]]
• ∀ (x) [teacher (x) ^ ∃ (y) [student (y) → likes (y, x)]]

Question 22

Identify the correct translation into logical notation of the following assertion.

"Some boys in the class are taller than all the girls" 

Note : taller(x,y) is true if x is taller than y.

• (∃x) (boy(x) → (∀y) (girl(y) ∧ taller(x,y)))
• (∃x) (boy(x) ∧ (∀y) (girl(y) ∧ taller(x,y)))
• (∃x) (boy(x) → (∀y) (girl(y) → taller(x,y)))
• (∃x) (boy(x) ∧ (∀y) (girl(y) → taller(x,y)))

Question 23

The following propositional statement is 
(P → (Q v R)) → ((P ^ Q) → R)

• satisfiable but not valid
  

• valid
  

• a contradiction
  

• none of the above
  

Question 24

Which of the following is a valid first order formula ? (Here α and β are first order formulae with x as their only free variable)

• A
• B
• C
• D

Question 25

Consider the following formula a and its two interpretations I1 and I2 

Which of the following statements is true?

• I2 satisfies α, I1 does not
  

• Neither I2 nor I2 satisfies α
  

• Both I1 and I2 satisfy α
  

• I1 satisfies α, I2 does not
  

Question 26

Consider the following logic program P A(x) <- B(x, y), C(y) <- B(x,x) Which of the following first order sentences is equivalent to P?

• A
• B
• C
• D

Question 27

The following resolution rule is used in logic programming.

Derive clause (P ∨ Q) from clauses (P ∨ R), (Q ∨ ¬R) 

Which of the following statements related to this rule is FALSE?

• ((P ∨ R) ∧ (Q ∨ ¬R)) ⇒ (P ∨ Q) is logically valid

• (P ∨ Q) ⇒ ((P ∨ R)) ∧ (Q ∨ ¬R)) is logically valid

• (P ∨ Q) is satisfiable if and only if (P ∨ R) ∨ (Q ∨ ¬R) is satisfiable

• (P ∨ Q) ⇒ FALSE if and only if both P and Q are unsatisfiable

Question 28

"If X, then Y unless Z" is represented by which of the following formulae in propositional logic? ("¬" is negation "^" is conjunction, and "→" is implication)

• (X ^ ¬ Z) → Y

• (X ^ Y) → ¬ Z

• (X → (Y ^ ¬ Z)

• (X → Y(^ ¬ Z)

Question 29

Consider two well-formed formulas in prepositional logic. 

F1 : P ⇒ ¬P
F2 : ( P⇒¬P)∨(¬P⇒P)

Which of the following statements is correct? 

• F1 unsatisfiable, F2 is satisfiable

• F1 and F2 are both satisfiable

• F1 is unsatisfiable, F2 is valid 

• F1 is satisfiable, F2 is valid

Question 30

Let a, b, c, d be propositions. Assume that the equivalences a ↔ (b V-b) and b ↔ c hold. Then the truth value of the formula (a ∧ b) → (a ∧ c) ∨ d) is always

• True
• False
• Same as the truth value of b
• Same as the truth value of d