To construct a truth table for the compound statement [tex]\(\sim p \wedge q\)[/tex], we need to examine all possible truth values for [tex]\(p\)[/tex] and [tex]\(q\)[/tex] and determine the truth value of [tex]\(\sim p \wedge q\)[/tex] in each case.
Let's examine each row of the truth table step by step:
1. Row 1:
- [tex]\(p = T\)[/tex]
- [tex]\(q = T\)[/tex]
- [tex]\(\sim p = F\)[/tex] (since [tex]\(p\)[/tex] is True, not [tex]\(p\)[/tex] is False)
- [tex]\(\sim p \wedge q = F \wedge T = F\)[/tex] (since the conjunction of False and True is False)
2. Row 2:
- [tex]\(p = T\)[/tex]
- [tex]\(q = F\)[/tex]
- [tex]\(\sim p = F\)[/tex] (since [tex]\(p\)[/tex] is True, not [tex]\(p\)[/tex] is False)
- [tex]\(\sim p \wedge q = F \wedge F = F\)[/tex] (since the conjunction of False and False is False)
3. Row 3:
- [tex]\(p = F\)[/tex]
- [tex]\(q = T\)[/tex]
- [tex]\(\sim p = T\)[/tex] (since [tex]\(p\)[/tex] is False, not [tex]\(p\)[/tex] is True)
- [tex]\(\sim p \wedge q = T \wedge T = T\)[/tex] (since the conjunction of True and True is True)
4. Row 4:
- [tex]\(p = F\)[/tex]
- [tex]\(q = F\)[/tex]
- [tex]\(\sim p = T\)[/tex] (since [tex]\(p\)[/tex] is False, not [tex]\(p\)[/tex] is True)
- [tex]\(\sim p \wedge q = T \wedge F = F\)[/tex] (since the conjunction of True and False is False)
Now, let's enter these values into the truth table:
[tex]\[
\begin{tabular}{|c|c|c|}
\hline
$p$ & $q$ & $\sim p \wedge q$ \\
\hline
T & T & F \\
\hline
T & F & F \\
\hline
F & T & T \\
\hline
F & F & F \\
\hline
\end{tabular}
\][/tex]
This completes the truth table for the compound statement [tex]\(\sim p \wedge q\)[/tex].
