![]() We always get confused while learning about the terms PCNF (Principal conjunction normal form) and CNF (Conjunction normal form), but there is a difference between both these terms. '.' is used to indicate that product is the main operator. The expression of PCNF with the help of these variables will be (X + Y' + Z). The expression of PCNF will be indicated in the following way:įor example: Suppose there are three variables X, Y, and Z. In other words, a formula ? will be known as the PCNF if ? is the product of max terms. The PCNF is used to perform the product of sums (POS). PCNF is also known as the Principal Conjunction normal form. Solution: Here, we will use X → Y ⇔ ¬X ∨ Y and De Morgan's law, and then we will get the following: (X ∧ Y ∧ Z) ∨ (X ∧ Y ∧ ¬Z) ∨ (¬X ∧ Y ∧ Z) ∨ (¬X ∧ ¬Y ∧ Z)Įxample 3: In this example, we have an expression X → ((X → Y) ∧ ¬(¬Y ∨ ¬X)), and we have to determine the PDNF without constructing the truth table. ⇔ (¬X ∧ Y) ∨ (¬X ∧ ¬Y) ∨ (X ∧ Y) Įxample 2: In this example, we have an expression (X ∧ Y) ∨ (¬X ∧ Z) ∨ (Y ∧ Z), and we have to determine the PDNF. If there are some identical minterms in the disjunction, then we will delete them.Įxample 1: In this example, we have an expression ¬X ∨ Y, and we have to determine the PDNF. After this, we can get minterms in disjunction with the help of introducing the missing factors. Now we will drop those elementary products, which are a contradiction.Now, we will apply negation on the variables with the help of De Morgan's laws, which will follow the distributive law.In this step, we will replace → with their equivalent formula, which is used to have ¬, ∧, and ∨.In the following way, we can obtain the PDNF of the given formula. (X ∧ Y ∧ Z) ∨ (X ∧ Y ∧ ¬Z) ∨ (¬X ∧ Y ∧ Z) ∨ (¬X ∧ ¬Y ∧ Z) Method 2: Without constructing the truth table So PDNF of (X ∧ Y) ∨ (¬X ∧ Z) ∨ (Y ∧ Z) will be: ![]() For the PDNF, we will only select the true values. In this table, there are various false values in the expression (X ∧ Y) ∨ (¬X ∧ Z) ∨ (Y ∧ Z). Solution: The truth table of X → Y is described as follows: In the last step, we will see the disjunction of these minterms, and it will be equivalent to the given formula.Įxample 1: In this example, we have an expression X → Y, and we have to determine the PDNF with the help of truth table.Now we will see the truth table, and for every truth value T, we will choose the minterm, which also has value T for the same combinations of T values of X and Y.In this step, we will first construct a truth table for the given formula.There are 2 methods by which we can obtain the PDNF of the given formula. The expression (X.Y'.Z) + (X'.Y.Z) + (X.Y.Z') can be known as an example of both DNF, and PDNF. ![]() The expression (X.Y'.Z) + (X'.Y.Z) + (X.Y) can be known as an example of DNF, but this expression cannot be PDNF.It is not necessary that the expression of DNF (Disjunction normal form) contains the same length of all the variables. We always get confused while learning about the terms PDNF (Principal disjunction normal form) and DNF (Disjunction normal form), but there is a huge difference between both these terms. ![]() + is used to indicate that sum is the main operator. The expression of PDNF with the help of these variables will be (X.Y'.Z) + (X'.Y.Z) + (X.Y.Z'). For example: Suppose there are three variables X, Y, and Z.
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