To determine the inverse of the given statement and evaluate its truth value, let's proceed with a step-by-step analysis.
1. Given Statements:
- Original statement: A number is negative if and only if it is less than 0.
- Let [tex]\( p \)[/tex] represent the proposition "A number is negative."
- Let [tex]\( q \)[/tex] represent the proposition "A number is less than 0."
2. Inverse of the Statement:
- The original statement in logical terms can be written as [tex]\( q \rightarrow p \)[/tex].
- The inverse of [tex]\( q \rightarrow p \)[/tex] is [tex]\( \neg q \rightarrow \neg p \)[/tex].
- Where [tex]\( \neg q \)[/tex] means "A number is not less than 0" (i.e., the number is 0 or positive).
- And [tex]\( \neg p \)[/tex] means "A number is not negative" (i.e., the number is 0 or positive).
3. Truth Value of the Inverse Statement:
- We need to evaluate [tex]\(\neg q \rightarrow \neg p\)[/tex].
- When a number is not less than 0 (i.e., the number is 0 or positive), it is indeed not negative.
- Therefore, [tex]\(\neg q \rightarrow \neg p\)[/tex] is always true.
4. Evaluation of Possible Answers:
- The inverse of the statement [tex]\( q \rightarrow p \)[/tex] is [tex]\(\neg q \rightarrow \neg p \)[/tex].
- We found that this inverse statement is true.
Reviewing the provided options, the correct ones are:
- [tex]\(\sim q \rightarrow \sim p\)[/tex]
- The inverse of the statement is true.
Thus, the correct answers are:
[tex]\(5\)[/tex] and [tex]\(8\)[/tex].
