Given the statement "[tex]$p:$[/tex] A number is negative" and "[tex]$q:$[/tex] A number is less than 0," let's analyze and find the inverse of the given statement and determine its validity.
1. Original Statement and Relationship:
The original statement can be expressed using logical notation as: [tex]\( p \leftrightarrow q \)[/tex], meaning "A number is negative if and only if it is less than 0."
2. Inverse of the Statement:
The inverse of a logical statement [tex]\( p \leftrightarrow q \)[/tex] is given by negating [tex]\( p \)[/tex] and [tex]\( q \)[/tex] individually and then forming the implication [tex]\( \neg q \rightarrow \neg p \)[/tex], which reads as: "If a number is not less than 0, then it is not negative."
3. Determine the Truth of the Inverse Statement:
- [tex]\(\neg q\)[/tex]: "A number is not less than 0." This means the number is greater than or equal to 0.
- [tex]\(\neg p\)[/tex]: "A number is not negative." This means the number is greater than or equal to 0.
When a number is not less than 0 (i.e., it is greater than or equal to 0), it cannot be negative. Thus, the inverse statement indeed holds true because a number that is not less than 0 (either 0 or positive) is not negative. Therefore, the inverse statement is true.
4. Conclusion:
The inverse of the statement "[tex]$p \leftrightarrow q$[/tex]" is "[tex]$\neg q \rightarrow \neg p$[/tex]," which translates to "If a number is not less than 0, then it is not negative."
Given the analysis, the correct answer is:
- The inverse of the statement is true.
Thus, the statement that represents the inverse correctly is [tex]\(\neg q \rightarrow \neg p\)[/tex], and the correct option is:
- The inverse of the statement is true.
