Question 7

Complete the truth table for the statement $A \wedge(B \vee C)$.
\begin{tabular}{|c|c|c|r|}
\hline $\mathrm{A}$ & $\mathrm{B}$ & $\mathrm{C}$ & $\mathrm{A} \wedge(\mathrm{B} \vee \mathrm{C})$ \\
\hline $\mathrm{T}$ & $\mathrm{T}$ & $\mathrm{T}$ & $? \mathrm{~V}$ \\
\hline $\mathrm{T}$ & $\mathrm{T}$ & $\mathrm{F}$ & $? \vee$ \\
\hline $\mathrm{T}$ & $\mathrm{F}$ & $\mathrm{T}$ & $? \mathrm{~V}$ \\
\hline $\mathrm{T}$ & $\mathrm{F}$ & $\mathrm{F}$ & $? \mathrm{~V}$ \\
\hline $\mathrm{F}$ & $\mathrm{T}$ & $\mathrm{T}$ & $? \vee$ \\
\hline $\mathrm{F}$ & $\mathrm{T}$ & $\mathrm{F}$ & $? \mathrm{~V}$ \\
\hline $\mathrm{F}$ & $\mathrm{F}$ & $\mathrm{T}$ & $? \mathrm{~V}$ \\
\hline $\mathrm{F}$ & $\mathrm{F}$ & $\mathrm{F}$ & ? \\
\hline
\end{tabular}

Question Help: $\triangle$ Message instructor

Step 1 ：The given logical statement is \(A \wedge (B \vee C)\). This statement is a conjunction of A and the disjunction of B and C. In other words, the statement is true if A is true and either B or C (or both) are true.

Step 2 ：We can use the truth values of A, B, and C given in the table and apply the logical operations.

Step 3 ：For the first row, A is True, B is True, and C is True. Since either B or C is True, the disjunction \(B \vee C\) is True. And since A is also True, the conjunction \(A \wedge (B \vee C)\) is True.

Step 4 ：For the second row, A is True, B is True, and C is False. Since B is True, the disjunction \(B \vee C\) is True. And since A is also True, the conjunction \(A \wedge (B \vee C)\) is True.

Step 5 ：For the third row, A is True, B is False, and C is True. Since C is True, the disjunction \(B \vee C\) is True. And since A is also True, the conjunction \(A \wedge (B \vee C)\) is True.

Step 6 ：For the fourth row, A is True, B is False, and C is False. Since neither B nor C is True, the disjunction \(B \vee C\) is False. And since A is True, the conjunction \(A \wedge (B \vee C)\) is False.

Step 7 ：For the remaining rows, A is False. Since the conjunction \(A \wedge (B \vee C)\) requires A to be True, the result is False regardless of the values of B and C.

Step 8 ：\(\boxed{\text{The completed truth table for the statement } A \wedge (B \vee C) \text{ is:}}\)

Step 9 ：\begin{tabular}{|c|c|c|r|} \hline A & B & C & A \wedge (B \vee C) \\ \hline T & T & T & T \\ \hline T & T & F & T \\ \hline T & F & T & T \\ \hline T & F & F & F \\ \hline F & T & T & F \\ \hline F & T & F & F \\ \hline F & F & T & F \\ \hline F & F & F & F \\ \hline \end{tabular}