To solve this problem, let's start by understanding the original statement and the logical symbols used.
### Original Statement:
"A number is negative if and only if it is less than 0."
### Logical Representation:
Let:
- \( p \) represent "A number is negative."
- \( q \) represent "A number is less than 0."
The original statement can be written in logical terms as:
[tex]\[ p \leftrightarrow q \][/tex]
This means "p if and only if q", or "A number is negative if and only if it is less than 0."
### Inverse of the Statement:
The inverse of a statement \( p \rightarrow q \) is \( \sim q \rightarrow \sim p \).
For our statement \( p \leftrightarrow q \), we consider the inverse of \( p \rightarrow q \) and its converse (since it's a biconditional statement), but primarily we need to find:
[tex]\[ \sim q \rightarrow \sim p \][/tex]
Where:
- \( \sim q \) means "A number is not less than 0." (i.e., the number is 0 or greater)
- \( \sim p \) means "A number is not negative." (i.e., the number is 0 or positive)
Thus, \( \sim q \rightarrow \sim p \) translates to:
"If a number is not less than 0, then it is not negative."
### Analyzing the Truth Value of the Inverse Statement:
To determine if this statement is true or false, let's consider what it means:
- \( \sim q \): A number is not less than 0 (i.e., it is 0 or greater).
- \( \sim p \): A number is not negative (i.e., it is 0 or positive).
It is clear that if a number is not less than 0 (it is 0 or positive), then it is not negative. Hence, the inverse statement holds true.
### Correct Answers:
Given this analysis, the correct answers are:
- \(\sim q \rightarrow \sim p\)
- The inverse of the statement is true.
So, the correct options are:
- \(\sim q \rightarrow \sim p\)
- The inverse of the statement is true.
The incorrect options are:
- The inverse of the statement is false.
- \( q \rightarrow p \)
- \(\sim p \leftrightarrow \sim q \)
- The inverse of the statement is sometimes true and sometimes false.
- [tex]\( q \leftrightarrow p \)[/tex]
