To prove the equivalence $(p \land \neg q) \lor (p \land q) = p$ using logical equivalence laws, let's go step by step, stating the laws applied at each step.

Expression:

$(p \land \neg q) \lor (p \land q) = p$

Step 1: Apply Distributive Law

The Distributive Law states that for any propositions $p$, $q$, and $r$, we have: $(p \land q) \lor (p \land r) \equiv p \land (q \lor r)$ We can apply this law to our expression: $(p \land \neg q) \lor (p \land q) = p \land (\neg q \lor q)$

Step 2: Apply Tautology Law

The Tautology Law tells us that: $q \lor \neg q \equiv \text{True}$ So, we simplify the expression further: $p \land (\neg q \lor q) = p \land \text{True}$

Step 3: Apply Identity Law

The Identity Law states: $p \land \text{True} \equiv p$ Thus, we simplify: $p \land \text{True} = p$

Final Conclusion:

$(p \land \neg q) \lor (p \land q) = p$

Conclusion:

We've successfully shown that $(p \land \neg q) \lor (p \land q)$ is logically equivalent to $p$, using the Distributive Law, Tautology Law, and Identity Law.

Would you like more details, or do you have any questions?

Related questions:

1. How can the Distributive Law be applied in other logical equivalences?
2. What is the Tautology Law, and why is it important in logical proofs?
3. How does the Identity Law help in simplifying logical expressions?
4. Can you prove $(p \lor q) \land (p \lor \neg q) = p$ using logical equivalence?
5. What are the implications of the Tautology Law in set theory?

Tip:

When proving logical equivalences, always look for opportunities to apply the basic laws like distribution, identity, and tautology for simplification.