Propositional and First Order Logic.

Question 81

In propositional logic P ↔ Q is equivalent to (Where ~ denotes NOT):

• ~( P ∨ Q ) ∧ ~ ( Q ∨ P )
• ( ~P ∨ Q ) ∧ (~ Q ∨ P )
• ( P ∨ Q ) ∧ ( Q ∨ P )
• ~( P ∨ Q ) → ~ ( Q ∨ P )

Question 82

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 83

Equivalent logical expression for the Well Formed Formula (WFF), ~(∀x) F[x]

• ∀x (~F[x])
• ~(∃x) F[x]
• ∃x (~F[x])
• ∀x F[x]

Question 84

The resolvent of the set of clauses (A ∨ B, ~A ∨ D, C ∨ ~B) is

• A ∨ B
• C ∨ D
• A ∨ C
• A ∨ D

Question 85

Consider the following English sentence : “Agra and Gwalior are both in India”. A student has written a logical sentence for the above English sentence in First-Order Logic using predicate In(x, y), which means x is in y, as follows : In(Agra, India) w In(Gwalior, India) Which one of the following is correct with respect to the above logical sentence ?

• It is syntactically valid but does not express the meaning of the English sentence.
• It is syntactically valid and expresses the meaning of the English sentence also.
• It is syntactically invalid but expresses the meaning of the English sentence.
• It is syntactically invalid and does not express the meaning of the English sentence.

Question 86

The equivalence of ¬∃x Q(x) is:

• ∃x ¬Q(x)
• ∀x ¬Q(x)
• ¬∃x ¬Q(x)
• ∀x Q(x)

Question 87

Which one of the following predicate formulae is NOT logically valid ? Note that W is a predicate formula without any free occurrence of x.

• ∀x(p(x) ∨ W) ≡ ∀x(px) ∨ W
• ∃x(p(x) ∧ W) ≡ ∃xp(x) ∧ W
• ∀x(p(x) → W) ≡ ∀xp(x) → W
• ∃x(p(x) → W) ≡ ∀xp(x) → W

Question 88

Match List-I with List-II List-I                          List-II (a) p → q                (i) ¬(q → ¬p) (b) p v q                   (ii) p ∧ ¬q (c) p ∧ q                  (iii) ¬p → q (d) ¬(p → q)            (iv) ¬p v q Choose the correct option from those given below:

• a-ii,b-iii,c-i,d-iv
• a-ii,b-i,c-iii,d-iv
• a-iv,b-i,c-iii,d-ii
• a-iv,b-iii,c-i,d-ii

Question 89

Which of the following is the principal conjunctive normal form for [(pVq) ∧ ~p → ~q]?

• pV~q
• pVq
• ~pVq
• ~pV~q