Let's complete the truth table for the inverse of a conditional statement step-by-step.
1. Determine [tex]\( p \rightarrow q \)[/tex]:
- For [tex]\( p = T \)[/tex] and [tex]\( q = T \)[/tex], [tex]\( p \rightarrow q = T \)[/tex]
- For [tex]\( p = T \)[/tex] and [tex]\( q = F \)[/tex], [tex]\( p \rightarrow q = F \)[/tex]
- For [tex]\( p = F \)[/tex] and [tex]\( q = T \)[/tex], [tex]\( p \rightarrow q = T \)[/tex]
- For [tex]\( p = F \)[/tex] and [tex]\( q = F \)[/tex], [tex]\( p \rightarrow q = T \)[/tex]
2. Determine [tex]\( \sim p \rightarrow \sim q \)[/tex]:
- For [tex]\( p = T \)[/tex] and [tex]\( q = T \)[/tex], [tex]\( \sim p = F \)[/tex] and [tex]\( \sim q = F \)[/tex]. [tex]\( \sim p \rightarrow \sim q = T \)[/tex]
- For [tex]\( p = T \)[/tex] and [tex]\( q = F \)[/tex], [tex]\( \sim p = F \)[/tex] and [tex]\( \sim q = T \)[/tex]. [tex]\( \sim p \rightarrow \sim q = T \)[/tex]
- For [tex]\( p = F \)[/tex] and [tex]\( q = T \)[/tex], [tex]\( \sim p = T \)[/tex] and [tex]\( \sim q = F \)[/tex]. [tex]\( \sim p \rightarrow \sim q = F \)[/tex]
- For [tex]\( p = F \)[/tex] and [tex]\( q = F \)[/tex], [tex]\( \sim p = T \)[/tex] and [tex]\( \sim q = T \)[/tex]. [tex]\( \sim p \rightarrow \sim q = T \)[/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 \hline [tex]$T$[/tex] & [tex]$T$[/tex] & [tex]$T$[/tex] & [tex]$T$[/tex] \\
\hline \hline [tex]$T$[/tex] & [tex]$F$[/tex] & [tex]$F$[/tex] & [tex]$T$[/tex] \\
\hline \hline [tex]$F$[/tex] & [tex]$T$[/tex] & [tex]$T$[/tex] & [tex]$F$[/tex] \\
\hline \hline [tex]$F$[/tex] & [tex]$F$[/tex] & [tex]$T$[/tex] & [tex]$T$[/tex] \\
\hline
\end{tabular}
