Find PDNF by constructing its PCNF of (Q ^ V P) -> (~ V R) ^ (~P V ~Q).

The given expression is (Q ^ V P) -> (~ V R) ^ (~P V ~Q). To find the PDNF (Product of Disjunctive Normal Form) by constructing its PCNF (Product of Conjunctive Normal Form), we need to follow the steps below:

1. Convert the given expression into its logical equivalent form without implication (->):

   The logical equivalent of A -> B is ~A V B. So, the given expression can be rewritten as:

   ~(Q ^ V P) V ((~ V R) ^ (~P V ~Q))

2. Apply De Morgan's Law:

   De Morgan's Law states that ~(A ^ B) is equivalent to ~A V ~B and ~(A V B) is equivalent to ~A ^ ~B. Applying this law, we get:

   (~Q V ~V P) V ((~ V R) ^ (~P V ~Q))

3. Convert the expression into CNF (Conjunctive Normal Form):

   CNF is a conjunction of one or more clauses, where a clause is a disjunction of literals. So, we need to distribute the OR over AND:

   ((~Q V ~V P) V (~ V R)) ^ ((~Q V ~V P) V (~P V ~Q))

4. Convert the CNF into DNF (Disjunctive Normal Form):

   DNF is a disjunction of one or more clauses, where a clause is a conjunction of literals. So, we need to distribute the AND over OR:

   ((~Q ^ ~V P) ^ (~ V R)) V ((~Q ^ ~V P) ^ (~P ^ ~Q))

This is the PDNF of the given expression.

Here is a table showing the steps and the transformations:

Step	Transformation	Expression
1	Remove implication	~(Q ^ V P) V ((~ V R) ^ (~P V ~Q))
2	Apply De Morgan's Law	(~Q V ~V P) V ((~ V R) ^ (~P V ~Q))
3	Convert to CNF	((~Q V ~V P) V (~ V R)) ^ ((~Q V ~V P) V (~P V ~Q))
4	Convert to DNF	((~Q ^ ~V P) ^ (~ V R)) V ((~Q ^ ~V P) ^ (~P ^ ~Q))

Please note that the symbols used in the expression are logical operators:

• ^ : AND
• V : OR
• ~ : NOT
• -> : IMPLIES