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.