Let's analyze and solve the given problems based on the uploaded image:

Problem 1: Proving Logical Equivalence

We have to prove the following statements:

A.

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

Let's use logical equivalence steps:

1. Apply De Morgan's Law: $\neg (p \land \neg q) = \neg p \lor \neg (\neg q) = \neg p \lor q$ Thus, the expression becomes: $(\neg p \lor q) \lor (\neg p \land \neg q)$
2. Apply the Absorption Law: $\neg p \lor (\neg p \land \neg q) = \neg p$ Therefore, $\neg (p \land \neg q) \lor (\neg p \land \neg q) \equiv \neg p$ This verifies that statement A is correct.

B.

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

Let's simplify:

1. Apply distribution: $\neg p \land (q \lor \neg q) = \neg p \land \text{True} = \neg p$ Thus, the original expression simplifies to: $\neg (\neg p) \lor (P \lor Q) \equiv p \lor (P \lor Q)$ By the idempotent law and identity laws, we get: $p \lor P \equiv P$ This verifies that statement B is also correct.

Problem 2: Simplifying Logical Statements

A.

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

Let's simplify it step-by-step:

1. Apply De Morgan's Law: $\neg (\neg p \land q) = p \lor \neg q$ The expression becomes: $(p \lor \neg q) \land (p \lor q)$
2. Apply distribution: $p \lor (\neg q \land q) = p \lor \text{False} = p$ Hence, the entire expression simplifies to: $p$

B.

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

Let's simplify:

1. Apply De Morgan's Law: $\neg (p \land q) = \neg p \lor \neg q$ The expression becomes: $(p \lor \neg q) \land (\neg p \lor \neg q)$
2. Apply distribution: $(p \land \neg p) \lor (p \land \neg q) \lor (\neg q \land \neg q)$ The simplified expression is: $p \land \neg q$

Final Results

1. For Problem 1:

   • A: $\neg (p \land \neg q) \lor (\neg p \land \neg q) \equiv \neg p$
   • B: $\neg ((\neg p \land q) \lor (\neg p \land \neg q)) \lor (P \lor Q) \equiv P$

2. For Problem 2:

   • A: $\neg (\neg p \land q) \land (p \lor q) \equiv p$
   • B: $(p \lor \neg q) \land \neg (p \land q) \equiv p \land \neg q$

Would you like further details or explanations on any part?

5 Questions for Further Practice

1. What is De Morgan's law, and how is it used in logical equivalence?
2. Can you apply the absorption law to simplify other logical expressions?
3. How does the distribution law work in logical equivalence?
4. Can two different expressions be equivalent without using basic logical equivalences?
5. How does using truth tables help verify logical equivalences?

Tip

When simplifying logical statements, always apply De Morgan's Law first to eliminate negations, making the remaining expressions easier to handle!