https://file.aiquickdraw.com/mathbot/file/1725795884335ofyjvzm2.jpg

It seems the text extraction from the image was unclear. However, based on what is visible from the image itself, the logical expression provided is:

\[
(p \land (p \lor \neg q)) \iff (p \land (\top \lor \neg q))
\]

You are asking for the **Principal Conjunctive Normal Form (PCNF)** and **Principal Disjunctive Normal Form (PDNF)**.

### Steps:
1. **PDNF (Disjunctive Normal Form)**: This form expresses the function as a disjunction of conjunctions (a sum of products).
   
2. **PCNF (Conjunctive Normal Form)**: This form expresses the function as a conjunction of disjunctions (a product of sums).

I will now calculate the PDNF and PCNF for this expression.The expression \((p \land (p \lor \neg q))\) simplifies logically to just \(p\). This means both the **Principal Disjunctive Normal Form (PDNF)** and the **Principal Conjunctive Normal Form (PCNF)** for this expression are simply:

\[
\text{PDNF} = p
\]
\[
\text{PCNF} = p
\]

Let me know if you'd like further details on the simplification process or any other questions. 

Here are 5 related questions:
1. What is the difference between DNF and CNF in Boolean logic?
2. How can you express a more complex expression using PDNF?
3. What are the advantages of using PDNF and PCNF in logical expressions?
4. How does logical simplification impact computational logic design?
5. Can PDNF or PCNF be useful in designing logic circuits?

**Tip:** Simplifying logical expressions before converting them into PDNF or PCNF often reduces the complexity of the final result.

Find PCNF and PDNF for the expression (p ∧ (p ∨ ¬q)) ⇔ (p ∧ (⊤ ∨ ¬q)).

Learn how to find the Principal Conjunctive Normal Form (PCNF) and Principal Disjunctive Normal Form (PDNF) of the expression (p ∧ (p ∨ ¬q)) ⇔ (p ∧ (⊤ ∨ ¬q)). This problem involves simplifying Boolean logic expressions and understanding CNF and DNF formats. Suitable for students studying discrete mathematics at the college level.

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