To find the correct answers, we need to understand the original statement and its inverse. Let's break this down step-by-step:
1. Original Statement:
- "A number is negative if and only if it is less than 0."
- This means [tex]\( p \leftrightarrow q \)[/tex].
- Where [tex]\( p \)[/tex]: A number is negative.
- And [tex]\( q \)[/tex]: A number is less than 0.
2. Inverse of the Statement:
- The inverse of a statement [tex]\( p \leftrightarrow q \)[/tex] is [tex]\( \sim q \leftrightarrow \sim p \)[/tex].
- Here, [tex]\( \sim p \)[/tex]: A number is not negative.
- And [tex]\( \sim q \)[/tex]: A number is not less than 0 (i.e., it is 0 or positive).
3. Evaluating the Inverse:
- Let's break down [tex]\( \sim q \leftrightarrow \sim p \)[/tex]:
[tex]\( \sim q \)[/tex]: A number is not less than 0.
[tex]\( \sim p \)[/tex]: A number is not negative.
- So, the inverse statement [tex]\( \sim q \leftrightarrow \sim p \)[/tex] translates to:
"A number is not less than 0 if and only if it is not negative."
- This statement is true, because:
If a number is not less than 0 (i.e., it’s 0 or positive), then it is not negative.
If a number is not negative, then it is not less than 0 (i.e., it’s 0 or positive).
So, the inverse statement [tex]\( \sim q \leftrightarrow \sim p \)[/tex] is true.
4. Summary of Options:
- The inverse of the statement is true. (Correct)
- [tex]\( q \leftrightarrow p \)[/tex]. (Incorrect) – This is the original statement.
- [tex]\( \sim p \leftrightarrow \sim q \)[/tex]. (Correct) – This is the correct representation of the inverse.
- The inverse of the statement is sometimes true and sometimes false. (Incorrect) – The inverse is always true, not sometimes.
- [tex]\( \sim q \rightarrow \sim p \)[/tex]. (Incorrect) – This represents only part of the inverse statement; it is not the full biconditional inverse.
- [tex]\( q \rightarrow p \)[/tex]. (Incorrect) – This is part of the original statement, not the inverse.
- The inverse of the statement is false. (Incorrect) – The inverse is true, not false.
Therefore, the correct answers are:
- The inverse of the statement is true.
- [tex]\( \sim p \leftrightarrow \sim q \)[/tex].
