Propositional and First Order Logic.

Question 11

P and Q are two propositions. Which of the following logical expressions are equivalent? 

• Only I, II and III
  

• Only I, II and IV
  

• All of I, II, III and IV
  

• Only I and II 
  

Question 12

Let Graph(x) be a predicate which denotes that x is a graph. Let Connected(x) be a predicate which denotes that x is connected. Which of the following first order logic sentences DOES NOT represent the statement: “Not every graph is connected”?

• A
• B
• C
• D

Question 13

Which of the following is TRUE about formulae in Conjunctive Normal Form?

• For any formula, there is a truth assignment for which at least half the clauses evaluate to true.

• For any formula, there is a truth assignment for which all the clauses evaluate to true

• There is a formula such that for each truth assignment, at most one-fourth of the clauses evaluate to true.

• None of the above

Question 14

Which one of the following propositional logic formulas is TRUE when exactly two of p, q, and r are TRUE?

• A
• B
• C
• D

Question 15

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

• C
  

• D
  

• B
  

• A
  

Question 16

The CORRECT formula for the sentence, “not all rainy days are cold” is

• A
• B
• C
• D

Question 17

Which one of the first order predicate calculus statements given below correctly express the following 
English statement? 

Tigers and lions attack if they are hungry or threatened. 

• B
  

• A
  

• C
  

• D
  

Question 18

Consider the following propositional statements: P1 : ((A ∧ B) → C)) ≡ ((A → C) ∧ (B → C)) P2 : ((A ∨ B) → C)) ≡ ((A → C) ∨ (B → C)) Which one of the following is true?

• P1 is a tautology, but not P2

• P2 is a tautology, but not P1

• P1 and P2 are both tautologies

• Both P1 and P2 are not tautologies

Question 19

A logical binary relation □ ,is defined as follows: 

Let ~ be the unary negation (NOT) operator, with higher precedence than □. 

Which one of the following is equivalent to A∧B ? 

(A) (~A □ B)  
(B) ~(A □ ~B) 
(C) ~(~A □ ~B)   
(D) ~(~A □ B) 

• C
  

• D
  

• B
  

• A
  

Question 20

Let P, Q and R be three atomic prepositional assertions. Let X denote (P v Q) → R and Y denote (P → R) v (Q → R). Which one of the following is a tautology?

• X ≡ Y
• X → Y
• Y → X
• ¬ Y → X