To determine if the given logical equivalence is true, we will simplify both sides of the equation.

The given equation is: $\neg(p \lor q) \land \neg(p \land q) \equiv \neg(p \lor q)$

We'll simplify both sides step by step.

Left Side: $\neg(p \lor q) \land \neg(p \land q)$

1. Distribute $\neg$ (NOT) using De Morgan's laws: $\neg(p \lor q) = \neg p \land \neg q$ $\neg(p \land q) = \neg p \lor \neg q$

2. Substitute these back into the left side: $(\neg p \land \neg q) \land (\neg p \lor \neg q)$

3. Distribute $\land$ (AND): $(\neg p \land \neg q \land \neg p) \lor (\neg p \land \neg q \land \neg q)$

   Simplify to: $\neg p \land \neg q$

So, the left side simplifies to: $\neg p \land \neg q$

Right Side: $\neg(p \lor q)$

Using De Morgan's laws, the right side simplifies to: $\neg(p \lor q) = \neg p \land \neg q$

Comparison

Since both sides simplify to $\neg p \land \neg q$, the given equivalence is true.

Conclusion

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

This logical equivalence holds true.

Would you like more details or have any questions?

Here are 8 related questions to explore further:

1. What is De Morgan's law in logic?
2. How do you simplify logical expressions using truth tables?
3. What is the difference between conjunction and disjunction in logic?
4. Can you show an example of simplifying a complex logical expression?
5. How do you use logical equivalences to prove logical statements?
6. What are some common logical fallacies and how can they be avoided?
7. How do you represent logical expressions in set theory?
8. What are the applications of Boolean algebra in computer science?

Tip: When working with logical expressions, always remember to apply De Morgan's laws and distribution rules accurately to simplify the expressions correctly.