Sure, let's complete the truth table step-by-step.
We need to fill in the column for the inverse of the conditional statement ([tex]\(\sim p \rightarrow \sim q\)[/tex]).
Here is the truth table format that we need to fill in:
\begin{tabular}{|c|c|c|c|}
\hline
p & [tex]$q$[/tex] & [tex]$p \rightarrow q$[/tex] & [tex]$\sim p \rightarrow \sim q$[/tex] \\
\hline
T & T & T & [ ] \\
\hline
T & F & F & [ ] \\
\hline
F & T & T & [ ] \\
\hline
F & F & T & [ ] \\
\hline
\end{tabular}
1. For the case when [tex]\(p\)[/tex] is True (T) and [tex]\(q\)[/tex] is True (T):
- [tex]\(\sim p\)[/tex] (not [tex]\(p\)[/tex]) is False (F) and [tex]\(\sim q\)[/tex] (not [tex]\(q\)[/tex]) is False (F)
- Thus, [tex]\(\sim p \rightarrow \sim q\)[/tex] is True (T)
2. For the case when [tex]\(p\)[/tex] is True (T) and [tex]\(q\)[/tex] is False (F):
- [tex]\(\sim p\)[/tex] (not [tex]\(p\)[/tex]) is False (F) and [tex]\(\sim q\)[/tex] (not [tex]\(q\)[/tex]) is True (T)
- Thus, [tex]\(\sim p \rightarrow \sim q\)[/tex] is True (T)
3. For the case when [tex]\(p\)[/tex] is False (F) and [tex]\(q\)[/tex] is True (T):
- [tex]\(\sim p\)[/tex] (not [tex]\(p\)[/tex]) is True (T) and [tex]\(\sim q\)[/tex] (not [tex]\(q\)[/tex]) is False (F)
- Thus, [tex]\(\sim p \rightarrow \sim q\)[/tex] is False (F)
4. For the case when [tex]\(p\)[/tex] is False (F) and [tex]\(q\)[/tex] is False (F):
- [tex]\(\sim p\)[/tex] (not [tex]\(p\)[/tex]) is True (T) and [tex]\(\sim q\)[/tex] (not [tex]\(q\)[/tex]) is True (T)
- Thus, [tex]\(\sim p \rightarrow \sim q\)[/tex] is True (T)
So, the completed truth table is:
\begin{tabular}{|c|c|c|c|}
\hline
p & [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}
Therefore, filling in the blanks we get:
1. For [tex]$p = T$[/tex] and [tex]$q = T$[/tex], [tex]\(\sim p \rightarrow \sim q\)[/tex] = T
2. For [tex]$p = T$[/tex] and [tex]$q = F$[/tex], [tex]\(\sim p \rightarrow \sim q\)[/tex] = T
3. For [tex]$p = F$[/tex] and [tex]$q = T$[/tex], [tex]\(\sim p \rightarrow \sim q\)[/tex] = F
4. For [tex]$p = F$[/tex] and [tex]$q = F$[/tex], [tex]\(\sim p \rightarrow \sim q\)[/tex] = T
