To prove the logical equivalence $\neg \neg (q \land (q \lor \neg p)) \equiv q$, we can follow these steps:

1. Double Negation Law: $\neg \neg A \equiv A$ Apply the double negation law to $\neg \neg (q \land (q \lor \neg p))$: $\neg \neg (q \land (q \lor \neg p)) \equiv q \land (q \lor \neg p)$

2. Distributive Law: $A \land (B \lor C) \equiv (A \land B) \lor (A \land C)$ Distribute $q$ over $q \lor \neg p$: $q \land (q \lor \neg p) \equiv (q \land q) \lor (q \land \neg p)$

3. Idempotent Law: $A \land A \equiv A$ Simplify $q \land q$ to $q$: $(q \land q) \lor (q \land \neg p) \equiv q \lor (q \land \neg p)$

4. Absorption Law: $A \lor (A \land B) \equiv A$ Apply the absorption law to $q \lor (q \land \neg p)$: $q \lor (q \land \neg p) \equiv q$

Thus, we have shown that: $\neg \neg (q \land (q \lor \neg p)) \equiv q$

Therefore, the logical equivalence $\neg \neg (q \land (q \lor \neg p)) \equiv q$ is proven.

Would you like more details or have any questions?

Here are 8 related questions that might interest you:

1. How do you prove other logical equivalences using truth tables?
2. What is the significance of De Morgan's laws in logical equivalences?
3. Can you explain the concept of tautology in logic?
4. How does the commutative property apply to logical operators?
5. What is the difference between a conjunction and a disjunction in logic?
6. How do you use the contrapositive in logical proofs?
7. Can you provide an example of using logical equivalences in a real-world scenario?
8. What are the primary laws of Boolean algebra used in simplifying logical expressions?

Tip: When simplifying logical expressions, always look for opportunities to apply basic logical equivalences like De Morgan's laws, the distributive law, and the absorption law to make the expression simpler and more manageable.