Sure, let's fill in the truth table step-by-step for the statement [tex]\(\sim d \rightarrow \sim c\)[/tex]:
1. List out all possible truth values for [tex]\(d\)[/tex] and [tex]\(c\)[/tex]:
- [tex]\(d = T\)[/tex], [tex]\(c = T\)[/tex]
- [tex]\(d = T\)[/tex], [tex]\(c = F\)[/tex]
- [tex]\(d = F\)[/tex], [tex]\(c = T\)[/tex]
- [tex]\(d = F\)[/tex], [tex]\(c = F\)[/tex]
2. Determine the values for [tex]\(\sim d\)[/tex] (not [tex]\(d\)[/tex]) and [tex]\(\sim c\)[/tex] (not [tex]\(c\)[/tex]) for each combination.
3. Evaluate the implication [tex]\(\sim d \rightarrow \sim c\)[/tex].
Here's the filled truth table:
[tex]\[
\renewcommand\arraystretch{1.5}
\begin{array}{|c|c|c|c|c|}
\hline
d & c & \sim d & \sim c & \sim d \rightarrow \sim c \\
\hline
T & T & F & F & T \\
\hline
T & F & F & T & T \\
\hline
F & T & T & F & F \\
\hline
F & F & T & T & T \\
\hline
\end{array}
\][/tex]
Explanation for each row:
1. [tex]\(d = T, c = T\)[/tex]:
- [tex]\(\sim d = F\)[/tex]
- [tex]\(\sim c = F\)[/tex]
- [tex]\(\sim d \rightarrow \sim c = F \rightarrow F\)[/tex]: This is true because an implication is true if the antecedent ([tex]\(\sim d\)[/tex]) is false or both the antecedent and the consequent ([tex]\(\sim c\)[/tex]) are false.
2. [tex]\(d = T, c = F\)[/tex]:
- [tex]\(\sim d = F\)[/tex]
- [tex]\(\sim c = T\)[/tex]
- [tex]\(\sim d \rightarrow \sim c = F \rightarrow T\)[/tex]: This is true because an implication is true if the antecedent ([tex]\(\sim d\)[/tex]) is false.
3. [tex]\(d = F, c = T\)[/tex]:
- [tex]\(\sim d = T\)[/tex]
- [tex]\(\sim c = F\)[/tex]
- [tex]\(\sim d \rightarrow \sim c = T \rightarrow F\)[/tex]: This is false because an implication is false if the antecedent ([tex]\(\sim d\)[/tex]) is true and the consequent ([tex]\(\sim c\)[/tex]) is false.
4. [tex]\(d = F, c = F\)[/tex]:
- [tex]\(\sim d = T\)[/tex]
- [tex]\(\sim c = T\)[/tex]
- [tex]\(\sim d \rightarrow \sim c = T \rightarrow T\)[/tex]: This is true because an implication is true if both the antecedent ([tex]\(\sim d\)[/tex]) and the consequent ([tex]\(\sim c\)[/tex]) are true.
And thus, the final truth table is as shown.
