Fill in the missing column of the following truth table.
\begin{tabular}{|c|c|c|c|}
\hline$p$ & $q$ & $\neg p$ & $\neg q$ \\
\hline$T$ & $T$ & $F$ & $F$ \\
\hline$F$ & $T$ & $T$ & $F$ \\
\hline$T$ & $F$ & $F$ & $T$ \\
\hline$F$ & $F$ & $T$ & $T$ \\
\hline
\end{tabular}
a)
\begin{tabular}{|c|}
\hline$\neg p \vee \neg q$ \\
\hline$F$ \\
\hline$T$ \\
\hline$T$ \\
\hline$T$ \\
\hline
\end{tabular}
b)
\begin{tabular}{|c|c|}
\hline$\rho \vee \neg q$ \\
\hline$\sigma$ & \\
\hline$T$ & \\
\hline$F$ & \\
\hline$T$ & \\
\hline
\end{tabular}
Answer (2)
The missing column for option a) ¬ p ∨ ¬ q has values F, T, T, T , while option b) p ∨ ¬ q has values T, F, T, T .
;
Evaluate ¬ p ∨ ¬ q : F ∨ F = F , T ∨ F = T , F ∨ T = T , T ∨ T = T .
Evaluate p ∨ ¬ q : T ∨ F = T , F ∨ F = F , T ∨ T = T , F ∨ T = T .
The missing column for option a) is F , T , T , T .
The missing column for option b) is T , F , T , T .
Explanation
Understanding the Problem We are given a truth table with columns for p , q , ¬ p , and ¬ q . We need to determine the correct values for the missing column for both options a) ¬ p ∨ ¬ q and b) p ∨ ¬ q .
Evaluating Option a For option a), we need to evaluate ¬ p ∨ ¬ q for each row of the truth table. Recall that A ∨ B is true if either A or B (or both) is true. Let's analyze each row:
Row 1: ¬ p = F , ¬ q = F . Thus, ¬ p ∨ ¬ q = F ∨ F = F .
Row 2: ¬ p = T , ¬ q = F . Thus, ¬ p ∨ ¬ q = T ∨ F = T .
Row 3: ¬ p = F , ¬ q = T . Thus, ¬ p ∨ ¬ q = F ∨ T = T .
Row 4: ¬ p = T , ¬ q = T . Thus, ¬ p ∨ ¬ q = T ∨ T = T .
So, the values for option a) are F , T , T , T .
Evaluating Option b For option b), we need to evaluate p ∨ ¬ q for each row of the truth table. Recall that A ∨ B is true if either A or B (or both) is true. Let's analyze each row:
Row 1: p = T , ¬ q = F . Thus, p ∨ ¬ q = T ∨ F = T .
Row 2: p = F , ¬ q = F . Thus, p ∨ ¬ q = F ∨ F = F .
Row 3: p = T , ¬ q = T . Thus, p ∨ ¬ q = T ∨ T = T .
Row 4: p = F , ¬ q = T . Thus, p ∨ ¬ q = F ∨ T = T .
So, the values for option b) are T , F , T , T .
Comparing the results Comparing our calculated values with the options provided, we see that option a) matches the given values F , T , T , T and option b) has values T , F , T , T .
Final Answer Therefore, the missing column for option a) is F , T , T , T and the missing column for option b) is T , F , T , T .
Examples
Truth tables are used in digital logic design to determine the output of a digital circuit for all possible combinations of inputs. For example, if p and q represent the inputs to a logic gate, the truth table can be used to determine the output of the gate for all possible input combinations. This is crucial in designing and troubleshooting digital circuits.
