Questions: Construct a truth table for the given compound statement. ¬(¬q ∧ ¬s) ∨(¬r ∨ ¬p) Fill in the truth table. p q r s ¬(¬q ∧ ¬s) ∨(¬r ∨ ¬p) T T T T T T T T F T T T F T T T T F F T T F T T V T F T F V T F F T V T F F F V

Construct a truth table for the given compound statement. ¬(¬q ∧ ¬s) ∨(¬r ∨ ¬p) Fill in the truth table. p q r s ¬(¬q ∧ ¬s) ∨(¬r ∨ ¬p) T T T T T T T T F T T T F T T T T F F T T F T T V T F T F V T F F T V T F F F V
Transcript text: Construct a truth table for the given compound statement. \[ \sim(\sim q \wedge \sim s) \vee(\sim r \vee \sim p) \] Fill in the truth table. \begin{tabular}{|c|c|c|c|c|} \hline $\mathbf{p}$ & $\mathbf{q}$ & $\mathbf{r}$ & $\mathbf{s}$ & $\sim(\sim \mathbf{q \wedge} \sim \mathbf{s}) \vee(\sim \mathbf{r} \vee \sim \mathbf{p})$ \\ \hline T & T & T & T & T \\ \hline T & T & T & F & T \\ \hline T & T & F & T & T \\ \hline T & T & F & F & T \\ \hline T & F & T & T & $\mathbf{V}$ \\ \hline T & F & T & F & $\mathbf{V}$ \\ \hline T & F & F & T & $\mathbf{V}$ \\ \hline T & F & F & F & $\mathbf{V}$ \\ \hline \end{tabular}
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Solution

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Solution Steps

Solution Approach

To construct a truth table for the given compound statement \(\sim(\sim q \wedge \sim s) \vee(\sim r \vee \sim p)\), we need to evaluate the expression for all possible truth values of the variables \(p\), \(q\), \(r\), and \(s\). This involves:

  1. Calculating the negations \(\sim q\), \(\sim s\), \(\sim r\), and \(\sim p\).
  2. Evaluating the conjunction \(\sim q \wedge \sim s\).
  3. Evaluating the negation of the conjunction \(\sim(\sim q \wedge \sim s)\).
  4. Evaluating the disjunction \(\sim r \vee \sim p\).
  5. Finally, evaluating the entire expression \(\sim(\sim q \wedge \sim s) \vee(\sim r \vee \sim p)\).
Step 1: Construct the Truth Table

To evaluate the compound statement \( \sim(\sim q \wedge \sim s) \vee(\sim r \vee \sim p) \), we first generate all possible combinations of truth values for the variables \( p \), \( q \), \( r \), and \( s \). Each variable can be either True (1) or False (0), leading to \( 2^4 = 16 \) combinations.

Step 2: Evaluate the Compound Statement

For each combination of truth values, we compute the following:

  • \( \sim q \)
  • \( \sim s \)
  • \( \sim r \)
  • \( \sim p \)
  • The conjunction \( \sim q \wedge \sim s \)
  • The negation \( \sim(\sim q \wedge \sim s) \)
  • The disjunction \( \sim r \vee \sim p \)
  • Finally, the entire expression \( \sim(\sim q \wedge \sim s) \vee(\sim r \vee \sim p) \)
Step 3: Compile Results

The results of the evaluations for each combination are as follows:

\[ \begin{array}{|c|c|c|c|c|} \hline p & q & r & s & \text{Result} \\ \hline 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 0 & 1 \\ 1 & 1 & 0 & 1 & 1 \\ 1 & 1 & 0 & 0 & 1 \\ 1 & 0 & 1 & 1 & 1 \\ 1 & 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 1 & 1 \\ 1 & 0 & 0 & 0 & 1 \\ 0 & 1 & 1 & 1 & 1 \\ 0 & 1 & 1 & 0 & 1 \\ 0 & 1 & 0 & 1 & 1 \\ 0 & 1 & 0 & 0 & 1 \\ 0 & 0 & 1 & 1 & 1 \\ 0 & 0 & 1 & 0 & 1 \\ 0 & 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 & 1 \\ \hline \end{array} \]

Final Answer

The final results indicate that the compound statement evaluates to True (1) for all combinations except when \( p = 1 \), \( q = 0 \), \( r = 1 \), and \( s = 0 \), where it evaluates to False (0). Thus, the overall truth table for the expression is complete.

The answer is boxed as follows: \[ \boxed{\text{Result is True for all combinations except } (1, 0, 1, 0)} \]

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