Normal and Principle Forms

1. Disjunctive Normal Forms (DNF) :
   A formula which is equivalent to a given formula and which consists of a sum of elementary products is called a disjunctive normal form of given formula.

   Example :
   (P ∧ ~ Q) ∨ (Q ∧ R) ∨ (~ P ∧ Q ∧~ R)

   • The DNF of formula is not unique.
2. Conjunctive Normal Form (CNF) :
   A formula which is equivalent to a given formula and which consists of a product of elementary sums is called a conjunctive normal form of given formula.

   Example :
   (P~ ∨ Q) ∧ (Q ∨ R) ∧ (~ P ∨ Q ∨ ~ R)

   • The CNF of formula is not unique.
   • If every elementary sum in CNF is tautology, then given formula is also tautology.
3. Principle Disjunctive Normal Form (PDNF) :
   An equivalent formula consisting of disjunctions of minterms only is called the principle disjunctive normal form of the formula.

   It is also known as sum-of-products canonical form.

   Example :
   (P ∧ ~ Q ∧ ~ R) ∨ (P ∧ ~ Q ∧ R) ∨ (~ P ∧ ~ Q ∧ ~ R)

   • The minterm consists of conjunctions in which each statement variable or its negation, but not both, appears only once.
   • The minterms are written down by including the variable if its truth value is T and its negation if its truth value is F.
4. Principle Conjunctive Normal Form (PCNF) :
   An equivalent formula consisting of conjunctions of maxterms only is called the principle conjunctive normal form of the formula.

   It is also known as product-of-sums canonical form.

   Example :
   (P ∨ ~ Q ∨ ~ R) ∧ (P ∨ ~ Q ∨ R) ∧ (~ P ∨ ~ Q ∨ ~ R)

   • The maxterm consists of disjunctions in which each variable or its negation, but not both, appears only once.
   • The dual of a minterm is called a maxterm.
   • Each of the maxterm has the truth value F for exactly one combination of the truth values of the variables.
   • The maxterms are written down by including the variable if its truth value is F and its negation if its truth value is T.