To analyze and solve this question, let's go through the problem step-by-step:
1. Understand the statement and its equivalent logical expressions:
- The original statement given is "A number is negative if and only if it is less than 0."
- We can denote the statements as:
- [tex]\( p \)[/tex]: A number is negative.
- [tex]\( q \)[/tex]: A number is less than 0.
2. Express the original statement in logical terms:
- The statement "A number is negative if and only if it is less than 0" translates to [tex]\( p \leftrightarrow q \)[/tex].
3. Determine the inverse of the original statement:
- The inverse of a statement [tex]\( p \rightarrow q \)[/tex] is [tex]\( \sim q \rightarrow \sim p \)[/tex].
- Since the original statement is [tex]\( p \leftrightarrow q \)[/tex], which can be expressed as [tex]\( (p \rightarrow q) \land (q \rightarrow p) \)[/tex], the inverse should apply to both implications:
- Consider [tex]\( p \rightarrow q \)[/tex]: The inverse is [tex]\( \sim q \rightarrow \sim p \)[/tex].
- Consider [tex]\( q \rightarrow p \)[/tex]: The inverse is [tex]\( \sim p \rightarrow \sim q \)[/tex] which is logically equivalent.
4. Check if the inverse statement is true:
- The inverse of "A number is negative if and only if it is less than 0" logically translates as [tex]\( \sim q \rightarrow \sim p \)[/tex].
- Here, [tex]\( \sim q \)[/tex] means "A number is not less than 0" (i.e., it is 0 or positive).
- And [tex]\( \sim p \)[/tex] means "A number is not negative" (i.e., it is 0 or positive).
- Thus, the inverse states, "If a number is not less than 0, then it is not negative," which is always true.
5. Evaluate the provided options:
- Option 1: [tex]\( \sim q \rightarrow \sim p \)[/tex] describes the inverse of [tex]\( p \rightarrow q \)[/tex] correctly.
- Option 2: indicates that the inverse of the statement is true.
- Option 3: states the inverse statement is false, which we know is incorrect.
- Option 4: [tex]\( q \leftrightarrow p \)[/tex] reaffirms the original statement, not the inverse.
- Option 5: [tex]\( q \rightarrow p \)[/tex] is not an inverse; it's another implication.
- Option 6: states the inverse is sometimes true and sometimes false, which is incorrect.
- Option 7: [tex]\( \sim p \leftrightarrow \sim q \)[/tex] is logically equivalent to the original statement's inverse.
Therefore, the correct answers are:
- [tex]\(\sim q \rightarrow \sim p\)[/tex]
- The inverse of the statement is true.
These correspond to options:
1. [tex]\(\sim q \rightarrow \sim p\)[/tex]
2. The inverse of the statement is true.
Thus, the correct answers are options 1 and 2.
