Question

Convert the following set of sentences to clausal form. For each step taken mention the law/operation used. We have several available laws

1. A ⇐⇒ (B ∨ E)

2. C ∧ F ⇒ ¬F

Answer 1

1) Given A (B V E)

A (B V E)

[A   (B V E)] [(B V E) A] { Law of DoubleImplies P Q = (P Q) (Q P)}

[¬A V (B V E)] [¬(B V E) V A] { Law of Implies (PQ) = ¬PVQ }

[¬A V (B V E)] [(¬B ¬E) V A] {By De Morgan's law ¬ (P V Q) = ¬P ¬Q }

[¬A V (E V B)] [(¬B ¬E) V A] {Commutative law (P V Q) = (Q V P)}

¬A V E V (B ¬B) ¬E V A {Associative law (P Q) R = P (Q R)}

¬A V E V (F) ¬E V A { we know that P ¬P = F }

¬A V (F) V E ¬E V A {Commutative law (P V Q) = (Q V P)}

¬A V (F) V (E ¬E) V A {Associative law (P Q) R = P (Q R)}

¬A V (F) V (F) V A { we know that P ¬P = F }

¬A V (F V F) V A {Associative law (P V Q) V R = PV (Q V R)}

¬A V (F) V A { we know that F V F = F }

¬A V (F) V A

Which is Required clausal form

2) Given C F ¬F

C F ¬F

¬ (C F) V ¬F { Law of Implies (PQ) = ¬PVQ }

(¬ C V ¬ F) V ¬F {By De Morgan's law ¬ (P Q) = ¬P V ¬Q }

¬ C V (¬ F V ¬F)   {Associative law (P V Q) V R = PV (Q V R)}

¬ C V (¬ F ) { we know that ¬F V ¬F = ¬F }

¬ C V ¬ ​​​​​​​F

Which is Required clausal form