Propositional and First Order Logic.

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Question 1

What is the correct translation of the following statement into mathematical logic? “Some real numbers are rational” 

 

  • A

  • B

  • C

  • D

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Question 2

Which one of the following is NOT logically equivalent to ¬∃x(∀y(α)∧∀z(β))?

  • ∀x(∃z(¬β)->∀y(α))

  • ∀x(∀z(β)->∃y(¬α))

  • ∀x(∀y(α)->∃z(¬β))

  • ∀x(∃y(¬α)->∃z(¬β))

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Question 3

What is the logical translation of the following statement? 

  "None of my friends are perfect."
gatecs201311

  • A

  • B

  • C

  • D

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Question 4

Which one of the following options is CORRECT given three positive integers x, y and z, and a predicate? 
 

        
      P(x) = ¬(x=1)∧∀y(∃z(x=y*z)⇒(y=x)∨(y=1))


 

  • P(x) being true means that x is a number other than 1
     

  • P(x) is always true irrespective of the value of x
     

  • P(x) being true means that x has exactly two factors other than 1 and x 
     

  • P(x) being true means that x is a prime number
     

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Question 5

Suppose the predicate F(x, y, t) is used to represent the statement that person x can fool person y at time t. which one of the statements below expresses best the meaning of the formula ∀x∃y∃t(¬F(x, y, t))?

  • Everyone can fool some person at some time

  • No one can fool everyone all the time

  • Everyone cannot fool some person all the time

  • No one can fool some person at some time

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Question 6

Which one of the following is the most appropriate logical formula to represent the statement? "Gold and silver ornaments are precious". The following notations are used: G(x): x is a gold ornament S(x): x is a silver ornament P(x): x is precious

  • ∀x(P(x)→(G(x)∧S(x)))

  • ∀x((G(x)∧S(x))→P(x))

  • ∃x((G(x)∧S(x))→P(x)

  • ∀x((G(x)∨S(x))→P(x))

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Question 7

The binary operation ? is defined as follows 

P
Q
P?Q
T
T
T
T
F
T
F
T
F
F
F
T


Which one of the following is equivalent to P∨Q? 
 

  • ¬Q?¬P

  • P?¬Q

  • ¬P?¬Q

  • ¬P?¬Q

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Question 8

CSE_2009_26

Which of the above two are equivalent?
 

  • II and III
     

  • I and IV
     

  • II and IV
     

  • I and III
     

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Question 9

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)

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Question 10

Let ν(x) mean x is a vegetarian, m(y) for y is meat, and e(x, y) for x eats y. Based on these, consider the following sentences : I. ∀x ν(x ) ⇔ (∀y e(x, y) ⇒ ¬m(y)) II. ∀x ν(x ) ⇔ (¬(∃y m(y) ∧e(x, y))) III. ∀x (∃y m(y) ∧e(x, y)) ⇔ ¬ν(x) One can determine that
  • Only I and II are equivalent sentences
  • Only II and III are equivalent sentences.
  • Only I and III are equivalent sentence .
  • I, II, and III are equivalent sentences.
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