To determine which statements are correct, let's proceed step-by-step:
1. Understanding the Original Statement:
- The original statement is: "A number is negative if and only if it is less than 0."
- In logical terms, this is [tex]\( p \leftrightarrow q \)[/tex], where:
- [tex]\( p \)[/tex]: A number is negative.
- [tex]\( q \)[/tex]: A number is less than 0.
2. Finding the Inverse:
- The inverse of the statement [tex]\( p \leftrightarrow q \)[/tex] is [tex]\( \sim p \leftrightarrow \sim q \)[/tex].
- [tex]\( \sim p \)[/tex]: A number is not negative.
- [tex]\( \sim q \)[/tex]: A number is not less than 0.
- Hence, the inverse statement is: "A number is not negative if and only if it is not less than 0."
- This can be restated as: "A number is non-negative if and only if it is zero or positive."
3. Evaluating the Truth of the Inverse:
- To determine whether the inverse statement is true, let's examine the conditions:
- If a number is non-negative ([tex]\(\sim p\)[/tex]), it must be either 0 or positive ([tex]\(\sim q\)[/tex]). This is true.
- If a number is zero or positive ([tex]\(\sim q\)[/tex]), it must be non-negative ([tex]\(\sim p\)[/tex]). This is also true.
Therefore, the inverse statement [tex]\( \sim p \leftrightarrow \sim q \)[/tex] is always true.
4. Matching the Correct Answers:
- "The inverse of the statement is true."
- [tex]\( \sim p \leftrightarrow \sim q \)[/tex]
Therefore, the correct answers to the given question are:
- [tex]\( \sim p \leftrightarrow \sim q \)[/tex]
- The inverse of the statement is true.
