Let's carefully work through the problem.
Given statement: A number is negative if and only if it is less than 0.
- p: A number is negative.
- q: A number is less than 0.
To represent the inverse of this statement:
1. The inverse of an implication [tex]\( p \rightarrow q \)[/tex] is [tex]\( \sim p \rightarrow \sim q \)[/tex].
- [tex]\( \sim p \)[/tex]: A number is not negative.
- [tex]\( \sim q \)[/tex]: A number is not less than 0 (i.e., a number is greater than or equal to 0).
Thus, the inverse statement is:
[tex]\[ \sim p \rightarrow \sim q \][/tex]
Now let's evaluate the truth of the inverse statement.
- When [tex]\( p \)[/tex] is false, [tex]\( \sim p \)[/tex] is true (the number is not negative).
- When [tex]\( q \)[/tex] is false, [tex]\( \sim q \)[/tex] is true (the number is not less than 0).
### Truth Evaluation for the Inverse Statement:
- [tex]\( \sim p \rightarrow \sim q \)[/tex] holds true when both [tex]\( \sim p \)[/tex] and [tex]\( \sim q \)[/tex] are true (i.e., the number is not negative and is greater than or equal to 0).
- [tex]\( \sim p \rightarrow \sim q \)[/tex] is false when [tex]\( \sim p \)[/tex] is true and [tex]\( \sim q \)[/tex] is false.
To summarize:
- The inverse of the statement [tex]\( \sim p \rightarrow \sim q \)[/tex] is true if both the number is not negative and the number is greater than or equal to 0 hold together.
- The inverse is false if any one of these conditions fails to hold when [tex]\( \sim p \)[/tex] is true and [tex]\( \sim q \)[/tex] is false.
Let's review the possible answers:
1. [tex]\( \sim p \leftrightarrow \sim q \)[/tex] - This is not the inverse of the original statement.
2. [tex]\( q \leftrightarrow p \)[/tex] - This is not the inverse of the original statement.
3. The inverse of the statement is sometimes true and sometimes false. - This is not correct because we determined that the inverse statement [tex]\( \sim p \rightarrow \sim q \)[/tex] can be evaluated as generally true under the given definitions.
4. The inverse of the statement is false. - This is incorrect; the logical interpretation shows it can be true.
5. [tex]\( \sim q \rightarrow \sim p \)[/tex] - This is incorrect.
6. The inverse of the statement is true. - This is correct.
7. [tex]\( q \rightarrow p \)[/tex] - This is not the inverse of the original statement.
Based on these evaluations, the correct answers are:
[tex]\[ \sim p \rightarrow \sim q \][/tex]
[tex]\[ \text{The inverse of the statement is true.} \][/tex]
Thus, the correct answers are:[tex]\[
\textbf{The inverse of the statement is true.}
\][/tex]
[tex]\[
\sim p \rightarrow \sim q
\][/tex]
