To complete the truth table for the logical statement [tex]\( p \wedge \sim q \)[/tex], we analyze each combination of truth values for [tex]\( p \)[/tex] and [tex]\( q \)[/tex]. Recall that [tex]\( \sim q \)[/tex] represents the negation of [tex]\( q \)[/tex], and [tex]\( p \wedge \sim q \)[/tex] represents the logical AND between [tex]\( p \)[/tex] and [tex]\( \sim q \)[/tex].
We will calculate the values step by step for each row:
1. For [tex]\( p = \text{True} \)[/tex] and [tex]\( q = \text{True} \)[/tex]:
- [tex]\( \sim q = \text{False} \)[/tex]
- [tex]\( p \wedge \sim q = \text{True} \wedge \text{False} = \text{False} \)[/tex]
2. For [tex]\( p = \text{True} \)[/tex] and [tex]\( q = \text{False} \)[/tex]:
- [tex]\( \sim q = \text{True} \)[/tex]
- [tex]\( p \wedge \sim q = \text{True} \wedge \text{True} = \text{True} \)[/tex]
3. For [tex]\( p = \text{False} \)[/tex] and [tex]\( q = \text{True} \)[/tex]:
- [tex]\( \sim q = \text{False} \)[/tex]
- [tex]\( p \wedge \sim q = \text{False} \wedge \text{False} = \text{False} \)[/tex]
4. For [tex]\( p = \text{False} \)[/tex] and [tex]\( q = \text{False} \)[/tex]:
- [tex]\( \sim q = \text{True} \)[/tex]
- [tex]\( p \wedge \sim q = \text{False} \wedge \text{True} = \text{False} \)[/tex]
Now we compile these results into a truth table:
[tex]\[
\begin{tabular}{|c|c|c|c|}
\hline
$p$ & $q$ & $\sim q$ & $p \wedge \sim q$ \\
\hline
\text{True} & \text{True} & \text{False} & \text{False} \\
\hline
\text{True} & \text{False} & \text{True} & \text{True} \\
\hline
\text{False} & \text{True} & \text{False} & \text{False} \\
\hline
\text{False} & \text{False} & \text{True} & \text{False} \\
\hline
\end{tabular}
\][/tex]
This table fully describes the logical statement [tex]\( p \wedge \sim q \)[/tex] for all possible truth values of [tex]\( p \)[/tex] and [tex]\( q \)[/tex].
