To find the Principal Disjunctive Normal Form (PDNF) and Principal Conjunctive Normal Form (PCNF) of the given logical expression $p \vee (p \rightarrow (q \vee (\neg q \rightarrow r)))$, we'll follow these steps:

1. Simplify the expression:

The expression given is $p \vee (p \rightarrow (q \vee (\neg q \rightarrow r)))$.

Step 1: Simplify the inner expression $\neg q \rightarrow r$.

$\neg q \rightarrow r \text{ is equivalent to } \neg \neg q \vee r, \text{ which simplifies to } q \vee r.$

So the expression becomes: $p \vee (p \rightarrow (q \vee (q \vee r)))$

Step 2: Simplify the expression $q \vee (q \vee r)$.

Since $q \vee q$ is just $q$, the expression simplifies to: $q \vee r$

Now the expression becomes: $p \vee (p \rightarrow (q \vee r))$

Step 3: Simplify $p \rightarrow (q \vee r)$.

$p \rightarrow (q \vee r)$ is equivalent to $\neg p \vee (q \vee r)$.

So the expression now is: $p \vee (\neg p \vee q \vee r)$

2. Simplify further:

The expression $p \vee (\neg p \vee q \vee r)$ can be simplified using the associative and commutative properties: $p \vee \neg p \vee q \vee r$

Since $p \vee \neg p$ is a tautology (always true), the expression simplifies to: $\text{True} \vee q \vee r \text{ which simplifies to True}$

3. Find the PDNF and PCNF:

Since the expression simplifies to True, both the PDNF and PCNF are trivial:

• PDNF (Principal Disjunctive Normal Form): True
• PCNF (Principal Conjunctive Normal Form): True

Summary

The PDNF and PCNF of the expression $p \vee (p \rightarrow (q \vee (\neg q \rightarrow r)))$ are both True.

Would you like to dive deeper into any part of this problem?

Here are 5 related questions for further exploration:

1. How would the PDNF and PCNF change if we modify the expression slightly, e.g., using $p \wedge (p \rightarrow (q \vee (\neg q \rightarrow r)))$?
2. Can you explain the difference between PDNF and PCNF in more detail?
3. What are tautologies and contradictions in logic, and how do they relate to PDNF and PCNF?
4. How would you approach finding the PDNF and PCNF of a more complex logical expression?
5. What is the significance of the associative and commutative properties in simplifying logical expressions?

Tip: When simplifying logical expressions, always look for tautologies like $p \vee \neg p$ or contradictions like $p \wedge \neg p$, as they can greatly simplify the process.