Questions: Which of the tables below is the truth table of the statement (p ∨ q) ∧ (¬p ∨ ¬r)?

Solution Steps

To determine the truth table for the statement \((p \vee q) \wedge (\sim p \vee \sim r)\), we need to evaluate the expression for all possible truth values of \(p\), \(q\), and \(r\). This involves creating a table with all combinations of truth values for \(p\), \(q\), and \(r\), and then computing the values of \(p \vee q\), \(\sim p\), \(\sim r\), \(\sim p \vee \sim r\), and finally \((p \vee q) \wedge (\sim p \vee \sim r)\).

1. List all possible combinations of truth values for \(p\), \(q\), and \(r\).
2. Compute the intermediate expressions \(p \vee q\), \(\sim p\), \(\sim r\), and \(\sim p \vee \sim r\).
3. Compute the final expression \((p \vee q) \wedge (\sim p \vee \sim r)\) for each combination.

We start by listing all possible combinations of truth values for the variables \(p\), \(q\), and \(r\). There are \(2^3 = 8\) combinations:

\[ \begin{array}{ccc} p & q & r \\ \hline \text{True} & \text{True} & \text{True} \\ \text{True} & \text{True} & \text{False} \\ \text{True} & \text{False} & \text{True} \\ \text{True} & \text{False} & \text{False} \\ \text{False} & \text{True} & \text{True} \\ \text{False} & \text{True} & \text{False} \\ \text{False} & \text{False} & \text{True} \\ \text{False} & \text{False} & \text{False} \\ \end{array} \]

For each combination, we compute the intermediate expressions \(p \vee q\), \(\neg p\), \(\neg r\), and \(\neg p \vee \neg r\):

\[ \begin{array}{cccccc} p & q & r & p \vee q & \neg p & \neg r & \neg p \vee \neg r \\ \hline \text{True} & \text{True} & \text{True} & \text{True} & \text{False} & \text{False} & \text{False} \\ \text{True} & \text{True} & \text{False} & \text{True} & \text{False} & \text{True} & \text{True} \\ \text{True} & \text{False} & \text{True} & \text{True} & \text{False} & \text{False} & \text{False} \\ \text{True} & \text{False} & \text{False} & \text{True} & \text{False} & \text{True} & \text{True} \\ \text{False} & \text{True} & \text{True} & \text{True} & \text{True} & \text{False} & \text{True} \\ \text{False} & \text{True} & \text{False} & \text{True} & \text{True} & \text{True} & \text{True} \\ \text{False} & \text{False} & \text{True} & \text{False} & \text{True} & \text{False} & \text{True} \\ \text{False} & \text{False} & \text{False} & \text{False} & \text{True} & \text{True} & \text{True} \\ \end{array} \]

Finally, we compute the value of the expression \((p \vee q) \wedge (\neg p \vee \neg r)\) for each combination:

\[ \begin{array}{cccc} p & q & r & (p \vee q) \wedge (\neg p \vee \neg r) \\ \hline \text{True} & \text{True} & \text{True} & \text{False} \\ \text{True} & \text{True} & \text{False} & \text{True} \\ \text{True} & \text{False} & \text{True} & \text{False} \\ \text{True} & \text{False} & \text{False} & \text{True} \\ \text{False} & \text{True} & \text{True} & \text{True} \\ \text{False} & \text{True} & \text{False} & \text{True} \\ \text{False} & \text{False} & \text{True} & \text{False} \\ \text{False} & \text{False} & \text{False} & \text{False} \\ \end{array} \]

Final Answer