Answered

Discover a world of knowledge at Westonci.ca, where experts and enthusiasts come together to answer your questions. Explore thousands of questions and answers from knowledgeable experts in various fields on our Q&A platform. Explore comprehensive solutions to your questions from knowledgeable professionals across various fields on our platform.

Complete the truth table for the inverse of a conditional statement.

Type the correct answer in each box. Use T for true and F for false.

\begin{tabular}{|c|c|c|c|}
\hline
[tex]$p$[/tex] & [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}

Sagot :

GinnyAnswer
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
Thanks for using our service. We're always here to provide accurate and up-to-date answers to all your queries. We appreciate your visit. Our platform is always here to offer accurate and reliable answers. Return anytime. Thank you for visiting Westonci.ca. Stay informed by coming back for more detailed answers.