Questions: Complete the truth table and determine whether or not ¬(p ∧ q) ≡ ¬p ∨ ¬q p q p ∧ q ¬(p ∧ q) ¬p ¬q --------------- T T F F F F T F F T F T F T T T T F F F T T T T Are the two statements equivalent? A. Yes, the columns are identical. B. No, the rows are not identical. C. Yes, the rows are identical. D. No, the columns are not identical.

Solution Steps

To determine whether \(\neg(p \wedge q) \equiv \neg p \vee \neg q\), we need to complete the truth table for both expressions and compare the results. If the columns for \(\neg(p \wedge q)\) and \(\neg p \vee \neg q\) are identical, then the statements are equivalent.

1. Create a truth table with all possible truth values for \(p\) and \(q\).
2. Calculate \(p \wedge q\) for each combination of \(p\) and \(q\).
3. Calculate \(\neg(p \wedge q)\) for each combination.
4. Calculate \(\neg p\) and \(\neg q\) for each combination.
5. Calculate \(\neg p \vee \neg q\) for each combination.
6. Compare the columns for \(\neg(p \wedge q)\) and \(\neg p \vee \neg q\).

We start by defining all possible truth values for \(p\) and \(q\): \[ \begin{array}{|c|c|} \hline p & q \\ \hline \text{True} & \text{True} \\ \text{True} & \text{False} \\ \text{False} & \text{True} \\ \text{False} & \text{False} \\ \hline \end{array} \]

Next, we calculate \(p \wedge q\) for each combination of \(p\) and \(q\): \[ \begin{array}{|c|c|c|} \hline p & q & p \wedge q \\ \hline \text{True} & \text{True} & \text{True} \\ \text{True} & \text{False} & \text{False} \\ \text{False} & \text{True} & \text{False} \\ \text{False} & \text{False} & \text{False} \\ \hline \end{array} \]

We then calculate \(\neg(p \wedge q)\) for each combination: \[ \begin{array}{|c|c|c|c|} \hline p & q & p \wedge q & \neg(p \wedge q) \\ \hline \text{True} & \text{True} & \text{True} & \text{False} \\ \text{True} & \text{False} & \text{False} & \text{True} \\ \text{False} & \text{True} & \text{False} & \text{True} \\ \text{False} & \text{False} & \text{False} & \text{True} \\ \hline \end{array} \]

Next, we calculate \(\neg p\) and \(\neg q\) for each combination: \[ \begin{array}{|c|c|c|c|} \hline p & q & \neg p & \neg q \\ \hline \text{True} & \text{True} & \text{False} & \text{False} \\ \text{True} & \text{False} & \text{False} & \text{True} \\ \text{False} & \text{True} & \text{True} & \text{False} \\ \text{False} & \text{False} & \text{True} & \text{True} \\ \hline \end{array} \]

We then calculate \(\neg p \vee \neg q\) for each combination: \[ \begin{array}{|c|c|c|c|c|} \hline p & q & \neg p & \neg q & \neg p \vee \neg q \\ \hline \text{True} & \text{True} & \text{False} & \text{False} & \text{False} \\ \text{True} & \text{False} & \text{False} & \text{True} & \text{True} \\ \text{False} & \text{True} & \text{True} & \text{False} & \text{True} \\ \text{False} & \text{False} & \text{True} & \text{True} & \text{True} \\ \hline \end{array} \]

Finally, we compare the columns for \(\neg(p \wedge q)\) and \(\neg p \vee \neg q\): \[ \begin{array}{|c|c|c|c|c|c|c|} \hline p & q & p \wedge q & \neg(p \wedge q) & \neg p & \neg q & \neg p \vee \neg q \\ \hline \text{True} & \text{True} & \text{True} & \text{False} & \text{False} & \text{False} & \text{False} \\ \text{True} & \text{False} & \text{False} & \text{True} & \text{False} & \text{True} & \text{True} \\ \text{False} & \text{True} & \text{False} & \text{True} & \text{True} & \text{False} & \text{True} \\ \text{False} & \text{False} & \text{False} & \text{True} & \text{True} & \text{True} & \text{True} \\ \hline \end{array} \]

Since the columns for \(\neg(p \wedge q)\) and \(\neg p \vee \neg q\) are identical, the statements are equivalent.

Final Answer