Let's complete the truth table for the inverse of a conditional statement step-by-step.
1. For the first row, where [tex]\( p \)[/tex] is True (T) and [tex]\( q \)[/tex] is True (T):
[tex]\[
\sim p \rightarrow \sim q = \sim (\text{True}) \rightarrow \sim (\text{True}) = \text{False} \rightarrow \text{False} = \text{True}
\][/tex]
2. For the second row, where [tex]\( p \)[/tex] is True (T) and [tex]\( q \)[/tex] is False (F):
[tex]\[
\sim p \rightarrow \sim q = \sim (\text{True}) \rightarrow \sim (\text{False}) = \text{False} \rightarrow \text{True} = \text{True}
\][/tex]
3. For the third row, where [tex]\( p \)[/tex] is False (F) and [tex]\( q \)[/tex] is True (T):
[tex]\[
\sim p \rightarrow \sim q = \sim (\text{False}) \rightarrow \sim (\text{True}) = \text{True} \rightarrow \text{False} = \text{False}
\][/tex]
4. For the fourth row, where [tex]\( p \)[/tex] is False (F) and [tex]\( q \)[/tex] is False (F):
[tex]\[
\sim p \rightarrow \sim q = \sim (\text{False}) \rightarrow \sim (\text{False}) = \text{True} \rightarrow \text{True} = \text{True}
\][/tex]
So, the completed truth table is:
\begin{tabular}{|c||c|c|c|}
\hline [tex]$p$[/tex] & [tex]$q$[/tex] & [tex]$p \rightarrow q$[/tex] & [tex]$\sim p \rightarrow \sim q$[/tex] \\
\hline T & T & T & T \\
\hline T & F & F & T \\
\hline F & T & T & F \\
\hline F & F & T & T \\
\hline
\end{tabular}
