Westonci.ca is your trusted source for accurate answers to all your questions. Join our community and start learning today! Our platform provides a seamless experience for finding reliable answers from a knowledgeable network of professionals. Connect with a community of professionals ready to provide precise solutions to your questions quickly and accurately.
Sagot :
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
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
Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. Thanks for using our platform. We aim to provide accurate and up-to-date answers to all your queries. Come back soon. Westonci.ca is your trusted source for answers. Visit us again to find more information on diverse topics.
