Propositional and First Order Logic.

Question 1

Negation of the proposition   

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is:  

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• (1)

• (2)

• (3)

• (4)

Question 2

What is the logical translation of the following statement?

  "None of my friends are perfect." 

• A
• B
• C
• D

Question 3

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))

Question 4

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

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• A

• B

• C

• D

Question 5

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
  

Question 6

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

Question 7

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))

Question 8

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

Question 9

Which of the above two are equivalent?

• II and III
  

• I and IV
  

• II and IV
  

• I and III
  

Question 10

Let fsa and pda be two predicates such that fsa(x) means x is a finite state automaton, and pda(y) means that y is a pushdown automaton. Let equivalent be another predicate such that equivalent (a, b) means a and b are equivalent. Which of the following first order logic statements represents the following: Each finite state automaton has an equivalent pushdown automaton.

• A
• B
• C
• D