Propositional and First Order Logic.

Question 61

In propositional logic if (P → Q) ∧ (R → S) and (P ∨ R) are two premises such that (P → Q) ∧ (R → S) P ∨ R Y Y is the premise :

• P ∨ R
• P ∨ S
• Q ∨ R
• Q ∨ S

Question 62

ECL is the fastest of all logic families. High speed in ECL is possible because transistors are used in difference amplifier configuration, in which they are never driven into ____.

• Race condition
• Saturation
• Delay
• High impedance

Question 63

Which of the following statements is true?

• The sentence S is a logical consequence of S₁,..., S_n if and only if S₁ ∧ S₂ ∧........ ∧ S_n → S is satisfiable.
• The sentence S is a logical consequence of S₁,..., Sn if and only if S₁ ∧ S₂ ∧........ ∧S_n → S is valid.
• The sentence S is a logical consequence of S₁,..., S_n if and only if S₁ ∧ S₂ ∧........ ∧ S_n ∧ S is consistent.
• The sentence S is a logical consequence of S₁,..., S_n if and only if S₁ ∧ S₂ ∧........ ∧ S_n ∧ S is inconsistent.

Question 64

The first order logic (FOL) statement ((R ∨ Q) ∧ (P ∨ ¬Q)) is equivalent to which of the following ?

• ((R ∨ ¬Q) ∧ (P ∨ ¬Q) ∧ (R ∨ P))
• ((R ∨ Q) ∧ (P ∨ ¬Q) ∧ (R ∨ P))
• ((R ∨ Q) ∧ (P ∨ ¬Q) ∧ (R ∨ ¬P))
• ((R ∨ Q) ∧ (P ∨ ¬Q) ∧ ( ¬R ∨ P))

Question 65

The Boolean function [~(~p ∧ q) ∧ ~( ~p ∧ ~q)] ∨ (p ∧ r)] is equal to the Boolean function:

• q
• p ∧ r
• p ∨ q
• p

Question 66

Let us assume that you construct ordered tree to represent the compound proposition (~ (p ∧ q)) ↔ (~ p ∨ ~ q). Then, the prefix expression and post-fix expression determined using this ordered tree are pgiven as ____ and _____ respectively.

• ↔~∧pq∨ ~ ~ pq, pq∧~p~q~∨↔
• ↔~∧pq∨ ~ p~q, pq∧~p~q~∨↔
• ↔~∧pq∨ ~ ~ pq, pq∧~p~~q∨↔
• ↔~∧pq∨ ~ p~ q, pq∧~p~ ~q∨↔

Question 67

Let A and B be sets in a finite universal set U. Given the following: |A – B|, |A ⊕ B|, |A| + |B| and |A ∪ B| Which of the following is in order of increasing size ?

• |A – B| < |A ⊕ B| < |A| + |B| < |A ∪ B|
  
• |A ⊕ B| < |A – B| < |A ∪ B| < |A| + |B|
  
• |A ⊕ B| < |A| + |B| < |A – B| < |A ∪ B|
  
• |A – B| < |A ⊕ B| < |A ∪ B| < |A| + |B|

Question 68

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.

Question 69

Consider the following logical inferences :

I1: If it is Sunday then school will not open. The school was open. 
Inference : It was not Sunday. 
 
I2: If it is Sunday then school will not open. It was not Sunday. 
Inference : The school was open. 

Which of the following is correct ?

• Both I₁ and I₂ are correct inferences.

• I₁ is correct but I2 is not a correct inference.

• I₁ is notcorrect but I2 is a correct inference.

• Both I₁ and I₂ are not correct inferences.

Question 70

Consider the statement, “Either – 2 ≤ x ≤ – 1 or 1 ≤ x ≤2”. The negation of this statement is

• x < – 2 or 2 < x or – 1 < x < 1
  
• x < – 2 or 2 < x
  
• – 1 < x < 1
  
• x ≤ – 2 or 2 < x or – 1 < x < 1