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Let's break down the statement \((\neg q \land \neg p) \land (\neg p \lor q)\) and fill in the truth table step by step.

### Variables:
- \( p \)
- \( q \)

### Negations:
- \( \neg p \): Negation of \( p \)
- \( \neg q \): Negation of \( q \)

### Intermediate operations:
- \( \neg q \land \neg p \): Logical AND between \( \neg q \) and \( \neg p \)
- \( \neg p \lor q \): Logical OR between \( \neg p \) and \( q \)

### Final expression:
- \( (\neg q \land \neg p) \land (\neg p \lor q) \): Logical AND between the two intermediate results.

### Step-by-step truth table completion:

| \( p \) | \( q \) | \( \neg p \) | \( \neg q \) | \( \neg q \land \neg p \) | \( \neg p \lor q \) | \( (\neg q \land \neg p) \land (\neg p \lor q) \) |
|--------|--------|-------------|-------------|--------------------------|--------------------|-------------------------------------------------|
|   T    |   T    |      F      |      F      |            F             |         F          |                       F                         |
|   T    |   F    |      F      |      T      |            F             |         F          |                       F                         |
|   F    |   T    |      T      |      F      |            F             |         T          |                       F                         |
|   F    |   F    |      T      |      T      |            T             |         T          |                       T                         |

### Explanation:
- **Row 1:** 
  - \( p = T \), \( q = T \), so \( \neg p = F \), \( \neg q = F \).
  - \( \neg q \land \neg p = F \land F = F \)
  - \( \neg p \lor q = F \lor T = F \)
  - \( F \land F = F \)

- **Row 2:**
  - \( p = T \), \( q = F \), so \( \neg p = F \), \( \neg q = T \).
  - \( \neg q \land \neg p = T \land F = F \)
  - \( \neg p \lor q = F \lor F = F \)
  - \( F \land F = F \)

- **Row 3:**
  - \( p = F \), \( q = T \), so \( \neg p = T \), \( \neg q = F \).
  - \( \neg q \land \neg p = F \land T = F \)
  - \( \neg p \lor q = T \lor T = T \)
  - \( F \land T = F \)

- **Row 4:**
  - \( p = F \), \( q = F \), so \( \neg p = T \), \( \neg q = T \).
  - \( \neg q \land \neg p = T \land T = T \)
  - \( \neg p \lor q = T \lor F = T \)
  - \( T \land T = T \)

Would you like any further details on this process, or do you have any questions?

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Here are 5 related questions to expand on this:
1. What is the purpose of constructing truth tables in logic?
2. How does the AND (\( \land \)) operation differ from the OR (\( \lor \)) operation?
3. Can you simplify the expression \( (\neg q \land \neg p) \land (\neg p \lor q) \)?
4. What is the significance of the final column in a truth table?
5. How can truth tables be used to check for logical equivalence between expressions?

**Tip:** When constructing truth tables, always start by evaluating the negations and then work step-by-step through logical operators (AND, OR) to avoid errors.

Construct a truth table for the statement (¬q ∧ ¬p) ∧ (¬p ∨ q).

Learn how to construct a truth table for the logical expression (¬q ∧ ¬p) ∧ (¬p ∨ q), breaking down negations, AND, and OR operations. Suitable for Grades 10-12.

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