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:

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?

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.