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Let's break down the given question into parts and solve them:

### Question 2:
#### a) **Obtain the Principal Disjunctive Normal Form (PDNF) of \((P \to Q) \land (Q \leftrightarrow R)\)**

1. **Step 1: Express the given logical expression in simpler terms:**
   - \( P \to Q \) is equivalent to \( \neg P \lor Q \) (Implication equivalence).
   - \( Q \leftrightarrow R \) is equivalent to \( (Q \land R) \lor (\neg Q \land \neg R) \) (Biconditional equivalence).

   Now the expression becomes:
   \[
   (\neg P \lor Q) \land [(Q \land R) \lor (\neg Q \land \neg R)]
   \]

2. **Step 2: Apply distributive laws:**
   To simplify this, we need to distribute the terms across the conjunction. We'll distribute \( \neg P \lor Q \) over the disjunction \( (Q \land R) \lor (\neg Q \land \neg R) \).

   \[
   (\neg P \lor Q) \land [(Q \land R) \lor (\neg Q \land \neg R)]
   \]

   Distribute this:
   \[
   [(\neg P \lor Q) \land (Q \land R)] \lor [(\neg P \lor Q) \land (\neg Q \land \neg R)]
   \]

3. **Step 3: Further simplify each part:**

   - The first part \((\neg P \lor Q) \land (Q \land R)\) can be expanded as:
     \[
     (\neg P \land Q \land R) \lor (Q \land R)
     \]
     This simplifies to \( (Q \land R) \), since \( Q \land R \) is common in both terms.
   
   - The second part \((\neg P \lor Q) \land (\neg Q \land \neg R)\) simplifies to:
     \[
     (\neg P \land \neg Q \land \neg R)
     \]
     because the term \( Q \) and \( \neg Q \) cannot both be true.

4. **Step 4: Final result:**
   The final disjunctive normal form (PDNF) is:
   \[
   (Q \land R) \lor (\neg P \land \neg Q \land \neg R)
   \]

#### b) **Show that \( \neg (P \lor Q) \) and \( \neg P \land \neg Q \) are logically equivalent.**

1. **Negation of disjunction:**
   - By **De Morgan's Law**, \( \neg (P \lor Q) \) is equivalent to:
     \[
     \neg P \land \neg Q
     \]

2. **Conclusion:**
   - Since both expressions \( \neg (P \lor Q) \) and \( \neg P \land \neg Q \) simplify to the same result, they are **logically equivalent**.

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Would you like a more detailed explanation or have any other questions on these topics?

Here are 5 related questions you could explore:
1. What is the Principal Conjunctive Normal Form (PCNF) of the same expression?
2. How can we prove De Morgan's laws using truth tables?
3. How do you convert any logical expression to its disjunctive normal form?
4. What is the difference between PDNF and PCNF in logic?
5. How do biconditional statements differ from simple conjunctions?

**Tip:** Always apply De Morgan's laws when dealing with negations of conjunctions or disjunctions—it helps in simplifying complex logical expressions efficiently.

a) Obtain the Principal disjunctive normal form of (P → Q) ∧ (Q ↔ R)
b) Show that ¬(P ∨ Q) and ¬P ∧ ¬Q are logically equivalent.

Learn how to find the Principal Disjunctive Normal Form (PDNF) for the expression (P → Q) ∧ (Q ↔ R) and prove the logical equivalence between ¬(P ∨ Q) and ¬P ∧ ¬Q using De Morgan's Law. Suitable for undergraduate students in logic or mathematics courses.

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