UGC NET Computer Science July 2018 - II | Question 76

Question 76

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76. Consider the following statements :
(a) False ⊨ True
(b) If α ⊨ (β ∧ γ) then α ⊨ β and α ⊨ γ.
Which of the following is correct with respect to the above statements ?

1. (1) Both statement (a) and statement (b) are false.
2. (2) Statement (a) is true but statement (b) is false.
3. (3) Statement (a) is false but statement (b) is true.
4. (4) Both statement (a) and statement (b) are true.

Answer : (4) Both statement (a) and statement (b) are true.

Explanation Question 76

• Statement (a) "False ⊨ True" is TRUE
  A ⊨ B means "A" logically entails "B" if and only if the sentence A ⇒ B is valid.
  We need to prove that False ⇒ True is valid.
  ∴ False ⇒ True is always true
  because we know : (a ⇒ b) ≡ (¬a ∨ b) So, False ⇒ True is equivalent to ( ¬Flase ∨ True ) which is always true.
   
  
• Statement (b) "if α ⊨ (β ∧ γ) then α ⊨ β and α ⊨ γ" is TRUE
  by constructing below truth table you will find out that
  " α ⊨ β and α ⊨ γ " are true for all the cases for which "α ⊨ (β and γ)" is true
   
  

  α	β	γ	(β ∧ γ)	α ⊨ (β ∧ γ)	α ⊨ β	α ⊨ γ
  				¬ α ∨ (β ∧ γ)	¬ α ∨ β	¬ α ∨ γ
  0	0	0	0	1	1	1
  0	0	1	0	1	1	1
  0	1	0	0	1	1	1
  0	1	1	1	1	1	1
  1	0	0	0	0	0	0
  1	0	1	0	0	0	1
  1	1	0	0	0	1	0
  1	1	1	1	1	1	1

Definition of Logical Entailment
A set of sentences (called premises) logically entails a sentence (called a conclusion) if and only if every truth assignment that satisfies the premises also satisfies the conclusion.

Example of Logical Entailment:
“KB logically entails S” if all the models that evaluate KB to True also evaluate S to True.
Denoted by: KB ⊨ S

Reference : Logical Entailment

Reference 1 : Realm world example of Logical Entailment

Reference 2 : https://www.ics.uci.edu/~welling/teaching/271fall09/HW6_sol.pdf#page=8

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