Let's break down the original statement and analyze its inverse.
### Original Statement
The given statement is:
"A number is negative if and only if it is less than 0."
This can be represented in logical terms as:
[tex]\[ p \leftrightarrow q \][/tex]
where,
- [tex]\( p \)[/tex]: A number is negative.
- [tex]\( q \)[/tex]: A number is less than 0.
### Inverse of the Statement
The inverse of a statement is obtained by negating both the hypothesis and the conclusion of the original statement. Thus, the inverse of [tex]\( p \leftrightarrow q \)[/tex] is:
[tex]\[ \sim q \rightarrow \sim p \][/tex]
where,
- [tex]\( \sim p \)[/tex]: A number is not negative.
- [tex]\( \sim q \)[/tex]: A number is not less than 0 (which means it is greater than or equal to 0).
### Analyzing the Inverse
We need to determine the truth value of the inverse statement [tex]\( \sim q \rightarrow \sim p \)[/tex].
1. [tex]\( \sim q \)[/tex] (A number is not less than 0) means the number is either 0 or positive.
2. If a number is 0 or positive, [tex]\( \sim p \)[/tex] (A number is not negative) is always true because 0 is non-negative and positive numbers are obviously non-negative.
Therefore, [tex]\( \sim q \rightarrow \sim p \)[/tex] is logically consistent and follows that the inverse statement should be evaluated as false.
So, let's evaluate the provided answers:
- The inverse of the statement is sometimes true and sometimes false: This is incorrect because the inverse is consistently false.
- [tex]\( q \leftrightarrow p \)[/tex]: This represents the original statement or its converse, not its inverse.
- [tex]\( \sim p \leftrightarrow \sim q \)[/tex]: This represents the contrapositive of the statement, not its inverse.
- The inverse of the statement is false: This is correct because the inverse is indeed false.
- [tex]\( q \rightarrow p \)[/tex]: This is not the inverse; it is the same as the original implication.
- The inverse of the statement is true: This is incorrect because we have established that the inverse is false.
- [tex]\( \sim q \rightarrow \sim p \)[/tex]: This correctly represents the inverse statement.
### Conclusion:
The correct answers are:
- The inverse of the statement is false.
- [tex]\( \sim q \rightarrow \sim p \)[/tex]
