Let's fill in the truth table step by step for each scenario:
1. When [tex]\( p \)[/tex] is True (T) and [tex]\( q \)[/tex] is True (T):
- Given that [tex]\( p \rightarrow q \)[/tex] is True (T).
- We need to find [tex]\( q \rightarrow p \)[/tex].
- Since [tex]\( q \rightarrow p \)[/tex] is also True (T), we fill in the box with [tex]\( T \)[/tex].
2. When [tex]\( p \)[/tex] is True (T) and [tex]\( q \)[/tex] is False (F):
- Given that [tex]\( p \rightarrow q \)[/tex] is False (F).
- We need to find [tex]\( q \rightarrow p \)[/tex].
- Since [tex]\( q \rightarrow p \)[/tex] is True (T), we fill in the box with [tex]\( T \)[/tex].
3. When [tex]\( p \)[/tex] is False (F) and [tex]\( q \)[/tex] is True (T):
- Given that [tex]\( p \rightarrow q \)[/tex] is True (T).
- We need to find [tex]\( q \rightarrow p \)[/tex].
- Since [tex]\( q \rightarrow p \)[/tex] is False (F), we fill in the box with [tex]\( F \)[/tex].
4. When [tex]\( p \)[/tex] is False (F) and [tex]\( q \)[/tex] is False (F):
- Given that [tex]\( p \rightarrow q \)[/tex] is True (T).
- We need to find [tex]\( q \rightarrow p \)[/tex].
- Since [tex]\( q \rightarrow p \)[/tex] is True (T), we fill in the box with [tex]\( T \)[/tex].
Thus, the completed truth table is:
\begin{tabular}{|c|c|c|c|}
\hline [tex]$p$[/tex] & [tex]$q$[/tex] & [tex]$p \rightarrow q$[/tex] & [tex]$q \rightarrow p$[/tex] \\
\hline T & T & T & T \\
\hline T & F & F & T \\
\hline F & T & T & F \\
\hline F & F & T & T \\
\hline
\end{tabular}
