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Complete the truth table for the given statement by filling in the required columns.

\begin{tabular}{|c|c|c|c|}
\hline
[tex]$p$[/tex] & [tex]$q$[/tex] & [tex]$\sim p$[/tex] & [tex]$\sim p \vee q$[/tex] \\
\hline
[tex]$T$[/tex] & [tex]$T$[/tex] & & \\
\hline
[tex]$T$[/tex] & [tex]$F$[/tex] & & \\
\hline
[tex]$F$[/tex] & [tex]$T$[/tex] & & \\
\hline
[tex]$F$[/tex] & [tex]$F$[/tex] & & \\
\hline
\end{tabular}

Complete the truth table.

\begin{tabular}{|c|c|c|c|}
\hline
[tex]$p$[/tex] & [tex]$q$[/tex] & [tex]$\sim p$[/tex] & [tex]$\sim p \vee q$[/tex] \\
\hline
[tex]$T$[/tex] & [tex]$T$[/tex] & [tex]$\nabla$[/tex] & [tex]$\nabla$[/tex] \\
\hline
[tex]$T$[/tex] & [tex]$F$[/tex] & [tex]$\nabla$[/tex] & [tex]$\nabla$[/tex] \\
\hline
[tex]$F$[/tex] & [tex]$T$[/tex] & [tex]$\nabla$[/tex] & [tex]$\nabla$[/tex] \\
\hline
[tex]$F$[/tex] & [tex]$F$[/tex] & [tex]$\nabla$[/tex] & [tex]$\nabla$[/tex] \\
\hline
\end{tabular}

Sagot :

GinnyAnswer
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]
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