2. Use truth tables to determine if the following propositions are equivalent.
[3] (a) ¬(p → ¬q) ⇔ p ∧ q
[3] (b) ¬p ∧ q ⇔ ¬(p → ¬q)

Let's analyze each part of the problem step by step, using truth tables to determine if the propositions are logically equivalent.

### (a) ¬(p → ¬q) ⇔ p ∧ q

1. **Construct truth table for each part:**
   - **p → ¬q**: This is the conditional statement "if p, then not q."
   - **¬(p → ¬q)**: This is the negation of "p → ¬q."
   - **p ∧ q**: This is the conjunction of "p" and "q."

The truth table for \( p, q, p \to \neg q, \neg (p \to \neg q), p \land q \):

| p   | q   | p → ¬q | ¬(p → ¬q) | p ∧ q |
|-----|-----|--------|------------|-------|
| T   | T   | F      | T          | T     |
| T   | F   | T      | F          | F     |
| F   | T   | T      | F          | F     |
| F   | F   | T      | F          | F     |

#### Analysis:
- For each truth value assignment to \( p \) and \( q \), the columns for \( ¬(p → ¬q) \) and \( p ∧ q \) match exactly.
- Therefore, \( \neg(p \to \neg q) \) is **logically equivalent** to \( p \land q \).

---

### (b) ¬p ∧ q ⇔ ¬(p → ¬q)

1. **Construct truth table for each part:**
   - **¬p ∧ q**: This is the conjunction of "not p" and "q."
   - **¬(p → ¬q)**: This is the negation of "p → ¬q" (from the previous part).

The truth table for \( p, q, \neg p, \neg p \land q, p \to \neg q, \neg (p \to \neg q) \):

| p   | q   | ¬p  | ¬p ∧ q | p → ¬q | ¬(p → ¬q) |
|-----|-----|-----|--------|--------|-----------|
| T   | T   | F   | F      | F      | T         |
| T   | F   | F   | F      | T      | F         |
| F   | T   | T   | T      | T      | F         |
| F   | F   | T   | F      | T      | F         |

#### Analysis:
- For each truth value assignment to \( p \) and \( q \), the columns for \( ¬p ∧ q \) and \( ¬(p → ¬q) \) match exactly.
- Therefore, \( \neg p \land q \) is **logically equivalent** to \( \neg(p \to \neg q) \).

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### Final Conclusion:
- (a) \( ¬(p → ¬q) ⇔ p ∧ q \) is true.
- (b) \( ¬p ∧ q ⇔ ¬(p → ¬q) \) is true.

Both propositions are logically equivalent as shown by the truth tables.

Would you like further details or clarification?

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Here are 5 related questions:
1. How can logical equivalence be proven without a truth table?
2. What are common logical equivalences involving conditionals and negations?
3. Can De Morgan’s laws help in simplifying logical propositions?
4. How does logical implication (\( \to \)) relate to conjunction (\( \land \)) and disjunction (\( \lor \))?
5. Why is negating a conditional statement tricky in logical reasoning?

**Tip:** When working with conditional statements, translating them into equivalent expressions can simplify complex logical propositions.

2. Use truth tables to determine if the following propositions are equivalent.
(a) ¬(p → ¬q) ⇔ p ∧ q
(b) ¬p ∧ q ⇔ ¬(p → ¬q)

Explore how to use truth tables to verify the logical equivalence of propositions such as ¬(p → ¬q) ⇔ p ∧ q and ¬p ∧ q ⇔ ¬(p → ¬q). This solution covers key logical concepts, including negation, conjunction, implication, and detailed truth table analysis. Suitable for students in Grades 10-12.

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