Sure, let's construct the truth table for the compound statement [tex]\( q \wedge (\sim p \vee r) \)[/tex]. We need to determine the truth value of this statement for all possible combinations of truth values of [tex]\( p \)[/tex], [tex]\( q \)[/tex], and [tex]\( r \)[/tex]. Here is the step-by-step construction of the truth table:
1. List all possible combinations of [tex]\( p \)[/tex], [tex]\( q \)[/tex], and [tex]\( r \)[/tex]:
There are 3 variables, so there are [tex]\(2^3 = 8\)[/tex] possible combinations of truth values.
[tex]\[
\begin{array}{|c|c|c|}
\hline
p & q & r \\
\hline
T & T & T \\
T & T & F \\
T & F & T \\
T & F & F \\
F & T & T \\
F & T & F \\
F & F & T \\
F & F & F \\
\hline
\end{array}
\][/tex]
2. Calculate [tex]\( \sim p \)[/tex]:
[tex]\[
\begin{array}{|c|c|c|c|}
\hline
p & q & r & \sim p \\
\hline
T & T & T & F \\
T & T & F & F \\
T & F & T & F \\
T & F & F & F \\
F & T & T & T \\
F & T & F & T \\
F & F & T & T \\
F & F & F & T \\
\hline
\end{array}
\][/tex]
3. Calculate [tex]\( \sim p \vee r \)[/tex]:
[tex]\[
\begin{array}{|c|c|c|c|c|}
\hline
p & q & r & \sim p & \sim p \vee r \\
\hline
T & T & T & F & T \\
T & T & F & F & F \\
T & F & T & F & T \\
T & F & F & F & F \\
F & T & T & T & T \\
F & T & F & T & T \\
F & F & T & T & T \\
F & F & F & T & T \\
\hline
\end{array}
\][/tex]
4. Calculate [tex]\( q \wedge (\sim p \vee r) \)[/tex]:
[tex]\[
\begin{array}{|c|c|c|c|c|c|}
\hline
p & q & r & \sim p & \sim p \vee r & q \wedge (\sim p \vee r) \\
\hline
T & T & T & F & T & T \\
T & T & F & F & F & F \\
T & F & T & F & T & F \\
T & F & F & F & F & F \\
F & T & T & T & T & T \\
F & T & F & T & T & T \\
F & F & T & T & T & F \\
F & F & F & T & T & F \\
\hline
\end{array}
\][/tex]
Here's the completed truth table for the compound statement [tex]\( q \wedge (\sim p \vee r) \)[/tex]:
[tex]\[
\begin{array}{|c|c|c|c|}
\hline
p & q & r & q \wedge (\sim p \vee r) \\
\hline
T & T & T & T \\
T & T & F & F \\
T & F & T & F \\
T & F & F & F \\
F & T & T & T \\
F & T & F & T \\
F & F & T & F \\
F & F & F & F \\
\hline
\end{array}
\][/tex]
This details each step to determine the truth values for the compound statement [tex]\( q \wedge (\sim p \vee r) \)[/tex].
