To solve this problem, let's analyze the given statement, find its inverse, and determine the truth value of that inverse. We have the following statement and its components:
- Original statement: "A number is negative if and only if it is less than 0."
- [tex]\( p \)[/tex]: A number is negative.
- [tex]\( q \)[/tex]: A number is less than 0.
This statement can be written in logical terms as [tex]\( p \leftrightarrow q \)[/tex] (p if and only if q).
### Finding the Inverse of the Statement
The inverse of a statement [tex]\( p \rightarrow q \)[/tex] is [tex]\( \sim p \rightarrow \sim q \)[/tex].
However, here we are dealing with [tex]\( p \leftrightarrow q \)[/tex], which is a biconditional statement.
The inverse of [tex]\( p \leftrightarrow q \)[/tex] is [tex]\( \sim q \rightarrow \sim p \)[/tex].
- [tex]\( \sim q \)[/tex]: A number is not less than 0 (i.e., a number is 0 or positive).
- [tex]\( \sim p \)[/tex]: A number is not negative (i.e., a number is 0 or positive).
Therefore, the inverse statement [tex]\( \sim q \rightarrow \sim p \)[/tex] reads: "If a number is not less than 0, then it is not negative."
### Evaluating the Truth Value of the Inverse
We need to determine if [tex]\( \sim q \rightarrow \sim p \)[/tex] is true or false.
Given:
- If [tex]\( \sim q \)[/tex] means the number is 0 or positive.
- If [tex]\( \sim p \)[/tex] means the number is 0 or positive.
Then the statement "If a number is not less than 0, then it is not negative" holds true in all cases:
- If a number is 0 or positive (not less than 0), it is indeed not negative.
Hence, [tex]\( \sim q \rightarrow \sim p \)[/tex] is true.
### Conclusion
Based on our analysis:
- The inverse of the statement is [tex]\( \sim q \rightarrow \sim p \)[/tex].
- The inverse statement is true.
Thus, the correct options to select are:
1. [tex]\( \sim q \rightarrow \sim p \)[/tex]
7. The inverse of the statement is true.
### Summary
The correct answers are:
[tex]\( 1. \sim q \rightarrow \sim p \)[/tex]
[tex]\( 7. The inverse of the statement is true \)[/tex]
