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Complete the truth table for the logical statement [tex]$p \wedge \sim q$[/tex].

\begin{tabular}{|c|c|c|c|}
\hline
[tex]$p$[/tex] & [tex]$q$[/tex] & [tex]$\sim q$[/tex] & [tex]$p \wedge \sim q$[/tex] \\
\hline
T & T & F & F \\
\hline
T & F & T & T \\
\hline
F & T & F & F \\
\hline
F & F & T & F \\
\hline
\end{tabular}



Answer :

GinnyAnswer
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].

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