simplify (𝒑 ∧ (𝒑 → 𝒓 ∨ 𝒒)) ∧ (𝒓 → 𝒒)

We can simplify the expression \( (p \land (p \to r \lor q)) \land (r \to q) \) using the basic rules of propositional logic.

### Step-by-step simplification:

1. **Implication rule**: Recall that \( p \to r \lor q \) is logically equivalent to \( \neg p \lor (r \lor q) \).
   
   So, we rewrite:
   \[
   p \land (\neg p \lor (r \lor q)) \land (r \to q)
   \]

2. **Implication rule**: Similarly, \( r \to q \) is equivalent to \( \neg r \lor q \).
   
   So now, the expression becomes:
   \[
   p \land (\neg p \lor (r \lor q)) \land (\neg r \lor q)
   \]

3. **Distributive property**: Distribute \( p \land \) across \( (\neg p \lor (r \lor q)) \):
   
   \[
   (p \land \neg p) \lor (p \land (r \lor q)) \land (\neg r \lor q)
   \]

4. **Tautology elimination**: The term \( p \land \neg p \) is a contradiction, so it simplifies to false (\( \bot \)).
   
   Now, the expression is:
   \[
   \bot \lor (p \land (r \lor q)) \land (\neg r \lor q)
   \]

   Which simplifies to:
   \[
   (p \land (r \lor q)) \land (\neg r \lor q)
   \]

5. **Distributive property**: Expand \( (p \land (r \lor q)) \land (\neg r \lor q) \):
   
   We can distribute the terms:
   \[
   p \land (r \lor q) \land (\neg r \lor q)
   \]

6. **Further simplification**:
   
   - Consider the inner terms \( (r \lor q) \land (\neg r \lor q) \). By distributive property:
     \[
     (r \land \neg r) \lor (r \land q) \lor (q \land \neg r) \lor (q \land q)
     \]
   - Simplifying this:
     \[
     \bot \lor (r \land q) \lor (q \land \neg r) \lor q
     \]
     This simplifies to:
     \[
     q \lor (r \land q) \lor (q \land \neg r)
     \]
     Which further simplifies to:
     \[
     q
     \]

   Now, we substitute this back:
   \[
   p \land q
   \]

### Final simplified expression:
\[
p \land q
\]

Would you like more details on any of these steps or have any other questions?

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Here are 5 related questions:
1. How does the implication rule work in logic?
2. What is the distributive property in logic?
3. Why does \( p \land \neg p \) simplify to false?
4. Can tautologies simplify more complex logical expressions?
5. How would adding another condition to the expression affect the simplification?

**Tip**: Remember that conjunctions with contradictions (\( p \land \neg p \)) or tautologies simplify very easily, helping reduce complex expressions quickly.

Explore how to simplify the propositional logic expression (p ∧ (p → r ∨ q)) ∧ (r → q) using logical equivalences, the implication rule, and the distributive property. Learn step-by-step simplification techniques suitable for undergraduate-level logic courses.

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