Sure, let's fill out the truth table step-by-step.
We'll look at rows where the variables [tex]\( p \)[/tex] and [tex]\( q \)[/tex] can both be either True (T) or False (F). For each combination, we'll need to determine the values of [tex]\( \sim p \)[/tex] (the negation of [tex]\( p \)[/tex]) and [tex]\( \sim p \vee q \)[/tex] (the logical OR between the negation of [tex]\( p \)[/tex] and [tex]\( q \)[/tex]).
Here is the detailed logic for each row:
1. First Row: [tex]\( p = T \)[/tex], [tex]\( q = T \)[/tex]
- [tex]\( \sim p \)[/tex]: Since [tex]\( p \)[/tex] is True (T), [tex]\( \sim p \)[/tex] (not [tex]\( p \)[/tex]) will be False (F).
- [tex]\( \sim p \vee q \)[/tex]: Since [tex]\( \sim p \)[/tex] is False (F) and [tex]\( q \)[/tex] is True (T), [tex]\( \sim p \vee q \)[/tex] will be True (T) because True OR False is True.
2. Second Row: [tex]\( p = T \)[/tex], [tex]\( q = F \)[/tex]
- [tex]\( \sim p \)[/tex]: Since [tex]\( p \)[/tex] is True (T), [tex]\( \sim p \)[/tex] will be False (F).
- [tex]\( \sim p \vee q \)[/tex]: Since [tex]\( \sim p \)[/tex] is False (F) and [tex]\( q \)[/tex] is False (F), [tex]\( \sim p \vee q \)[/tex] will be False (F) because False OR False is False.
3. Third Row: [tex]\( p = F \)[/tex], [tex]\( q = T \)[/tex]
- [tex]\( \sim p \)[/tex]: Since [tex]\( p \)[/tex] is False (F), [tex]\( \sim p \)[/tex] will be True (T).
- [tex]\( \sim p \vee q \)[/tex]: Since [tex]\( \sim p \)[/tex] is True (T) and [tex]\( q \)[/tex] is True (T), [tex]\( \sim p \vee q \)[/tex] will be True (T) because True OR True is True.
4. Fourth Row: [tex]\( p = F \)[/tex], [tex]\( q = F \)[/tex]
- [tex]\( \sim p \)[/tex]: Since [tex]\( p \)[/tex] is False (F), [tex]\( \sim p \)[/tex] will be True (T).
- [tex]\( \sim p \vee q \)[/tex]: Since [tex]\( \sim p \)[/tex] is True (T) and [tex]\( q \)[/tex] is False (F), [tex]\( \sim p \vee q \)[/tex] will be True (T) because True OR False is True.
Now, completing the truth table:
[tex]\[
\begin{array}{|c|c|c|c|}
\hline
p & q & \sim p & \sim p \vee q \\
\hline
T & T & F & T \\
\hline
T & F & F & F \\
\hline
F & T & T & T \\
\hline
F & F & T & T \\
\hline
\end{array}
\][/tex]
So, the completed truth table is:
[tex]\[
\begin{array}{|c|c|c|c|}
\hline
p & q & \sim p & \sim p \vee q \\
\hline
T & T & F & T \\
\hline
T & F & F & F \\
\hline
F & T & T & T \\
\hline
F & F & T & T \\
\hline
\end{array}
\][/tex]
