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The problem asks to simplify the following logical formulas using logical laws:

1. \[ (\neg p \land q) \to (q \to p) \land p \]
2. \[ (\neg p \to q) \to (\neg p \land q) \]

### Let's break them down one by one:

#### 1. \[ (\neg p \land q) \to (q \to p) \land p \]

We'll simplify this step by step:

- First, recall that \( a \to b \) is logically equivalent to \( \neg a \lor b \).
  Thus, the expression becomes:
  \[
  \neg (\neg p \land q) \lor ((q \to p) \land p)
  \]
- Using De Morgan's law on \( \neg (\neg p \land q) \), we get:
  \[
  (\neg \neg p \lor \neg q) \lor ((q \to p) \land p)
  \]
  which simplifies to:
  \[
  (p \lor \neg q) \lor ((\neg q \lor p) \land p)
  \]
- Now we distribute \( \lor \) over \( \land \):
  \[
  (p \lor \neg q) \lor ((\neg q \lor p) \land p)
  \]
  This can be simplified further by logical absorption, but the core simplification would be:
  \[
  (p \lor \neg q)
  \]
  Therefore, the simplified expression for the first formula is:
  \[
  p \lor \neg q
  \]

#### 2. \[ (\neg p \to q) \to (\neg p \land q) \]

- Using the same equivalence \( a \to b \equiv \neg a \lor b \), the expression becomes:
  \[
  \neg (\neg p \to q) \lor (\neg p \land q)
  \]
- Simplifying \( \neg (\neg p \to q) \), we first rewrite \( \neg p \to q \) as \( p \lor q \), so we get:
  \[
  \neg (p \lor q) \lor (\neg p \land q)
  \]
- Applying De Morgan’s law on \( \neg (p \lor q) \), we get:
  \[
  (\neg p \land \neg q) \lor (\neg p \land q)
  \]
- Now, distribute:
  \[
  \neg p \land (\neg q \lor q)
  \]
  Since \( \neg q \lor q \) is a tautology (always true), we simplify to:
  \[
  \neg p
  \]

Thus, the simplified form for the second expression is:
\[
\neg p
\]

### Final simplified results:
1. \( p \lor \neg q \)
2. \( \neg p \)

Would you like more details on the simplifications or have any questions?

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Here are five questions to expand on this topic:
1. How does De Morgan's law work in simplifying logical formulas?
2. What is the importance of tautologies in logic?
3. Can you explain how logical equivalences like \( a \to b \equiv \neg a \lor b \) are derived?
4. Why does distribution work differently in logic compared to arithmetic?
5. How do you prove a logical expression using truth tables?

**Tip**: When simplifying logical expressions, always remember key equivalences like \( \neg (p \land q) = \neg p \lor \neg q \) and \( \neg (p \lor q) = \neg p \land \neg q \) for De Morgan's laws.

Simplify the following logical formulas:
1. (¬p ∧ q) → (q → p) ∧ p
2. (¬p → q) → (¬p ∧ q)

Simplify logical formulas using De Morgan's Law and Logical Equivalences. Step-by-step solution provided for the expressions: (¬p ∧ q) → (q → p) ∧ p and (¬p → q) → (¬p ∧ q).

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