To solve this problem, we need to evaluate the logical statement [tex]\(\neg p \rightarrow \neg q\)[/tex] given that [tex]\(p\)[/tex] is true and [tex]\(q\)[/tex] is true. Let's break it down step-by-step.
1. Given values:
- [tex]\(p\)[/tex] is true.
- [tex]\(q\)[/tex] is true.
2. Evaluate [tex]\(\neg p\)[/tex]:
- Since [tex]\(p\)[/tex] is true, [tex]\(\neg p\)[/tex] (the negation of [tex]\(p\)[/tex]) is false.
3. Evaluate [tex]\(\neg q\)[/tex]:
- Since [tex]\(q\)[/tex] is true, [tex]\(\neg q\)[/tex] (the negation of [tex]\(q\)[/tex]) is false.
4. Evaluate the implication [tex]\(\neg p \rightarrow \neg q\)[/tex]:
- In logic, an implication [tex]\(A \rightarrow B\)[/tex] is true in all cases except when [tex]\(A\)[/tex] is true and [tex]\(B\)[/tex] is false.
- Here, [tex]\(A\)[/tex] is [tex]\(\neg p\)[/tex] and [tex]\(B\)[/tex] is [tex]\(\neg q\)[/tex].
- We have [tex]\(\neg p\)[/tex] as false and [tex]\(\neg q\)[/tex] as false.
5. Determine the truth value of [tex]\(\neg p \rightarrow \neg q\)[/tex]:
- When the antecedent ([tex]\(\neg p\)[/tex]) is false, the implication [tex]\(\neg p \rightarrow \neg q\)[/tex] is always true, regardless of the truth value of the consequent ([tex]\(\neg q\)[/tex]).
Hence, [tex]\({ }^{\sim} p \rightarrow{ }^{\sim} q\)[/tex] is true given that both [tex]\(p\)[/tex] and [tex]\(q\)[/tex] are true.
