To solve this problem, let's first understand the original statement and then determine its inverse. The original statement is:
"A number is negative if and only if it is less than 0."
This can be written in logical terms as:
[tex]\[ p \leftrightarrow q \][/tex]
where:
- \( p \): "A number is negative."
- \( q \): "A number is less than 0."
An inverse of a logical statement \( p \rightarrow q \) is:
[tex]\[ \sim p \rightarrow \sim q \][/tex]
(If not \( p \) then not \( q \)).
However, since our original statement is biconditional (\( p \leftrightarrow q \)), we take the inverse of one direction first. We understand that:
[tex]\[ p \rightarrow q \][/tex]
and its inverse is:
[tex]\[ \sim p \rightarrow \sim q \][/tex].
In the original statement \( p \rightarrow q \):
- \( p \): A number is negative.
- \( q \): A number is less than 0.
Inverse of \( p \rightarrow q \) becomes:
[tex]\[ \sim q \rightarrow \sim p \][/tex]
where:
- \( \sim q \): A number is not less than 0.
- \( \sim p \): A number is not negative.
So the inverse is:
"If a number is not less than 0, then it is not negative."
Now we need to determine the truth value of this inverse statement. Consider the possible cases:
- If a number is not less than 0, then it can be either zero or positive.
- A number that is zero is not negative.
- A number that is positive is not negative.
Thus, the inverse statement "If a number is not less than 0, then it is not negative" is true in the general case and false in special cases depending on the negation interpretation.
To answer fully, we need to identify the correct expressions and the truthfulness of the inverse statement. These are:
- \( \sim q \rightarrow \sim p \)
- The inverse of the statement is sometimes true and sometimes false.
The correct answers are:
- \( \sim q \rightarrow \sim p \)
- The inverse of the statement is sometimes true and sometimes false.
