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The expression in the image is asking to obtain the **Principal Disjunctive Normal Form (PDNF)** for the logical expression:

\[
\varphi = p \lor (\neg p \lor (\neg p \lor q)) 
\]

The **Principal Disjunctive Normal Form (PDNF)**, also known as the **Sum of Products** form, is a way to express logical formulas as a disjunction (OR) of conjunctions (ANDs), where each conjunction consists of literals (either a variable or its negation).

Let's simplify the given expression step by step:

1. **Original Expression**: 
   \[
   \varphi = p \lor (\neg p \lor (\neg p \lor q))
   \]

2. **Simplification**:
   \[
   \neg p \lor (\neg p \lor q)
   \]
   Using the associative property of disjunction, we simplify the parentheses:
   \[
   \neg p \lor \neg p \lor q \equiv \neg p \lor q
   \]
   
   Now, the entire expression becomes:
   \[
   p \lor (\neg p \lor q) \equiv p \lor \neg p \lor q
   \]
   
3. **Final Simplification**:
   \[
   p \lor \neg p \lor q
   \]
   Since \(p \lor \neg p\) is always true (a tautology), this simplifies to:
   \[
   \text{True} \lor q \equiv \text{True}
   \]

Therefore, the expression simplifies to **True**, meaning the PDNF for this particular formula is just "True."

Would you like more details on how PDNF is obtained or further clarification? Here are some additional questions to explore:

1. What are the steps to convert a more complex logical formula to PDNF?
2. How does PDNF relate to Karnaugh maps in digital logic design?
3. What are the key differences between PDNF and PCNF (Principal Conjunctive Normal Form)?
4. How do you derive PDNF for expressions involving more variables?
5. What are common tautologies and contradictions in logic simplifications?

**Tip**: Always simplify logical expressions by applying fundamental identities like De Morgan's Laws, tautologies, and contradictions early in the process to make the derivation of normal forms easier.

Obtain PDNF for the logical expression ϕ = p ∨ (¬p ∨ (¬p ∨ q))

This problem explores obtaining the Principal Disjunctive Normal Form (PDNF) for the logical expression ϕ = p ∨ (¬p ∨ (¬p ∨ q)). The solution involves simplifying the expression using key Boolean Algebra theorems and logical identities. Ideal for students studying logic or digital design.

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