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Transform the following into Conjunctive Normal Form (CNF)

The statements are given as follows:

  • P ∨ (~P & Q & R)

  • (~P & Q) ∨ (P & ~Q) & S


Conjunctive normal form implies that all terms in the equation must have the "&" symbol between them. In the two sentences given above, we can see we have the "∨" symbol between the terms. Let us convert each of them into conjunctive normal form one-by-one.




Statement 1: P ∨ (~P & Q & R)

P ∨ (~P & Q & R)

= (P ∨ ~P) & (P ∨ Q) & (P ∨ R)

= (P ∨ Q) & (P ∨ R)


We used two properties here:

  • While opening the brackets, whatever is outside the bracket must be multiplied/operated with each term inside the bracket.

  • (P ∨ ~P) = NULL, as it is always TRUE.


Statement 2: (~P & Q) ∨ (P & ~Q) & S

(~P & Q) ∨ (P & ~Q) & S

= ((~P ∨ (P & ~Q)) & (Q ∨ (P & ~Q))) & S

= ((~P ∨ P) & (~P ∨~Q)) & ((Q ∨ P) & (Q ∨~Q)) & S

= (~P ∨~Q) & (Q ∨ P) & S


We have converted both statements into Conjunctive Normal Form.

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