Let's analyze the given statements and what it means to find the inverse.
Firstly, let's define the original statements:
- [tex]\( p \)[/tex]: A number is negative.
- [tex]\( q \)[/tex]: A number is less than 0.
In logical notation, if we have a statement [tex]\( q \rightarrow p \)[/tex], it translates to:
"If a number is less than 0, then it is negative."
To find the inverse of this statement, we need to negate both the hypothesis and the conclusion, which gives us the statement [tex]\( \sim q \rightarrow \sim p \)[/tex].
- [tex]\( \sim q \)[/tex]: A number is not less than 0.
- [tex]\( \sim p \)[/tex]: A number is not negative (i.e., non-negative, which means the number is either zero or positive).
Thus, the inverse statement [tex]\( \sim q \rightarrow \sim p \)[/tex] translates to:
"If a number is not less than 0, then it is not negative."
Next, we need to determine whether this inverse statement is true or false.
In the inverse statement:
- If a number is not less than 0 (i.e., it is 0 or positive), then it must be not negative.
This statement holds true because a number that is either zero or positive is indeed not negative. Thus, the inverse of the statement is true.
### Correct Answers:
1. The inverse of the statement is true.
2. [tex]\( \sim q \rightarrow \sim p \)[/tex]
So, the correct selections from the provided options are:
- The inverse of the statement is true.
- [tex]\( \sim q \rightarrow \sim p \)[/tex]
These two options accurately describe the inverse of the original statement and its truth value.
