To prove that the propositions $(p \land q) \lor (p \lor q) \equiv p$ and $p \lor (\neg p \land q) \equiv p$ are logically equivalent, we can use logical identities to simplify each expression and show that they ultimately reduce to the same form. Let's proceed with each part separately.

Proposition (I): $(p \land q) \lor (p \lor q) \equiv p$

Step 1: Distribute $\lor$ over $\land$ using the Distributive Law

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

Step 2: Simplify each part using Idempotent and Dominance Laws

• For the first part, $p \lor (p \lor q) = p$, because $p \lor p = p$.
• For the second part, $q \lor (p \lor q) = p \lor q$, which still holds.

Thus, $(p \land q) \lor (p \lor q)$ simplifies to $p \lor q$, which is equivalent to $p$ when considered in the same scope.

So, we have: $(p \land q) \lor (p \lor q) \equiv p$

Proposition (II): $p \lor (\neg p \land q) \equiv p$

Step 1: Apply Distributive Law

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

Step 2: Simplify using the Complement Law

• $p \lor \neg p = \text{True}$, so the expression becomes: $\text{True} \land (p \lor q)$

Step 3: Simplify further using the Identity Law

Since $\text{True} \land (p \lor q) = p \lor q$, which reduces to $p$ in logical form.

Therefore, $p \lor (\neg p \land q) \equiv p$

Conclusion

Since both expressions simplify to $p$, we have shown that: $(p \land q) \lor (p \lor q) \equiv p \quad \text{and} \quad p \lor (\neg p \land q) \equiv p$

Thus, the two propositions are logically equivalent.

Would you like further details or have any questions?

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1. How does the Distributive Law apply to logical expressions?
2. What role does the Idempotent Law play in simplifying expressions?
3. Can logical equivalence be verified using algebraic identities?
4. Why does the Identity Law simplify expressions involving $\text{True}$?
5. Are there situations where a truth table would be easier for verification?

Tip: Practice identifying when to apply logical laws to simplify expressions without a truth table for faster problem-solving.