To solve this problem, we will first identify the statement given and then determine its inverse logically.
### Original Statement:
The statement given is:
"A number is negative if and only if it is less than 0."
Let's denote:
- [tex]\( p \)[/tex]: A number is negative.
- [tex]\( q \)[/tex]: A number is less than 0.
The original statement can be written as:
[tex]\[ p \leftrightarrow q \][/tex]
This denotes a biconditional statement meaning "p if and only if q." In other words, [tex]\( p \)[/tex] and [tex]\( q \)[/tex] are equivalent.
### Inverse of the Statement:
The inverse of a statement [tex]\(\rightarrow\)[/tex] is formed by negating both the hypothesis and the conclusion. For a biconditional statement [tex]\( p \leftrightarrow q \)[/tex], the inverse will be:
[tex]\[ \neg p \leftrightarrow \neg q \][/tex]
This reads as:
"A number is not negative if and only if it is not less than 0."
### Determine the Truth Value of the Inverse:
To check the truth value of the inverse, we consider whether the inverse statement holds in all conditions:
- The statement [tex]\( p \leftrightarrow q \)[/tex] means [tex]\( p \)[/tex] is true if and only if [tex]\( q \)[/tex] is true.
- If [tex]\( p \)[/tex] is true, then [tex]\( q \)[/tex] must be true and if [tex]\( p \)[/tex] is false, then [tex]\( q \)[/tex] must be false.
Therefore, the negation of both statements will also represent a true relationship:
- If a number is not negative (i.e., [tex]\(\neg p\)[/tex]), then it is not less than 0 (i.e., [tex]\(\neg q\)[/tex]).
Thus, the inverse statement [tex]\(\neg p \leftrightarrow \neg q\)[/tex] remains true.
### Summary of Correct Answers:
- The inverse of the statement is true.
- [tex]\(\sim p \leftrightarrow \sim q\)[/tex]
Given our analysis, the correct selections are:
- The inverse of the statement is true.
- [tex]\(\sim p \leftrightarrow \sim q\)[/tex]
The other options do not accurately describe the inverse statement or its truth value.
