Normal and Principle Forms

**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.

**Conjunctive Normal Form (CNF) :**

A formula which is equivalent to a given formula and which consists of a product of elementary products 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.

**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.

**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.