Answer :
Sure, let's carefully analyze each statement given the original premise "If a number is negative, the additive inverse is positive."
1. Original Statement:
- The original statement is "If a number is negative, the additive inverse is positive."
- Let's denote "a number is negative" by [tex]\( p \)[/tex] and "the additive inverse is positive" by [tex]\( q \)[/tex].
- Therefore, the original statement in logical notation is [tex]\( p \rightarrow q \)[/tex].
- This is indeed correct.
- Statement: If [tex]\( p = \)[/tex] "a number is negative" and [tex]\( q = \)[/tex] "the additive inverse is positive," the original statement is [tex]\( p \rightarrow q \)[/tex].
- Evaluation: True
2. Inverse Statement:
- The inverse of the original statement [tex]\( p \rightarrow q \)[/tex] is [tex]\( \neg p \rightarrow \neg q \)[/tex].
- In words, this means "If a number is not negative, the additive inverse is not positive."
- Statement: If [tex]\( p = \)[/tex] "a number is negative" and [tex]\( q = \)[/tex] "the additive inverse is positive," the inverse of the original statement is [tex]\( \neg p \rightarrow \neg q \)[/tex].
- Evaluation: True
3. Converse Statement:
- The converse of the original statement [tex]\( p \rightarrow q \)[/tex] is [tex]\( q \rightarrow p \)[/tex].
- In words, this means "If the additive inverse is positive, then the number is negative."
- Statement: If [tex]\( p = \)[/tex] "a number is negative" and [tex]\( q = \)[/tex] "the additive inverse is positive," the converse of the original statement is [tex]\( q \rightarrow p \)[/tex].
- Evaluation: True
4. Contrapositive Statement:
- The contrapositive of the original statement [tex]\( p \rightarrow q \)[/tex] is [tex]\( \neg q \rightarrow \neg p \)[/tex].
- In words, this means "If the additive inverse is not positive, the number is not negative."
- According to the statement provided, [tex]\( q \)[/tex] is redefined as "a number is negative" and [tex]\( p \)[/tex] as "the additive inverse is positive".
- This means the contrapositive should be formulated in terms of "If the additive inverse is not positive, then the number is not negative."
- Statement: If [tex]\( q = \)[/tex] "a number is negative" and [tex]\( p = \)[/tex] "the additive inverse is positive," the contrapositive of the original statement is [tex]\( \neg p \rightarrow \neg q \)[/tex].
- Evaluation: False
5. Wrong Converse Statement:
- The converse of the statement given is incorrectly framed: "If [tex]\( q = \)[/tex] 'a number is negative' and [tex]\( p = \)[/tex] 'the additive inverse is positive,' the converse of the original statement is [tex]\( q \rightarrow p \)[/tex]."
- However, [tex]\( q \rightarrow p \)[/tex] is correct for converse, but as per the question's presentation, this framing may have an error in syntax.
- Statement: If [tex]\( q = \)[/tex] "a number is negative" and [tex]\( p = \)[/tex] "the additive inverse is positive," the converse of the original statement is [tex]\( q \rightarrow p \)[/tex].
- Evaluation: False due to syntax or misrepresentation problems.
Thus, the true statements are:
1. If [tex]\( p = \)[/tex] "a number is negative" and [tex]\( q = \)[/tex] "the additive inverse is positive," the original statement is [tex]\( p \rightarrow q \)[/tex].
2. If [tex]\( p = \)[/tex] "a number is negative" and [tex]\( q = \)[/tex] "the additive inverse is positive," the inverse of the original statement is [tex]\( \neg p \rightarrow \neg q \)[/tex].
3. If [tex]\( p = \)[/tex] "a number is negative" and [tex]\( q = \)[/tex] "the additive inverse is positive," the converse of the original statement is [tex]\( q \rightarrow p \)[/tex].
True Indices: [1, 2, 3]
1. Original Statement:
- The original statement is "If a number is negative, the additive inverse is positive."
- Let's denote "a number is negative" by [tex]\( p \)[/tex] and "the additive inverse is positive" by [tex]\( q \)[/tex].
- Therefore, the original statement in logical notation is [tex]\( p \rightarrow q \)[/tex].
- This is indeed correct.
- Statement: If [tex]\( p = \)[/tex] "a number is negative" and [tex]\( q = \)[/tex] "the additive inverse is positive," the original statement is [tex]\( p \rightarrow q \)[/tex].
- Evaluation: True
2. Inverse Statement:
- The inverse of the original statement [tex]\( p \rightarrow q \)[/tex] is [tex]\( \neg p \rightarrow \neg q \)[/tex].
- In words, this means "If a number is not negative, the additive inverse is not positive."
- Statement: If [tex]\( p = \)[/tex] "a number is negative" and [tex]\( q = \)[/tex] "the additive inverse is positive," the inverse of the original statement is [tex]\( \neg p \rightarrow \neg q \)[/tex].
- Evaluation: True
3. Converse Statement:
- The converse of the original statement [tex]\( p \rightarrow q \)[/tex] is [tex]\( q \rightarrow p \)[/tex].
- In words, this means "If the additive inverse is positive, then the number is negative."
- Statement: If [tex]\( p = \)[/tex] "a number is negative" and [tex]\( q = \)[/tex] "the additive inverse is positive," the converse of the original statement is [tex]\( q \rightarrow p \)[/tex].
- Evaluation: True
4. Contrapositive Statement:
- The contrapositive of the original statement [tex]\( p \rightarrow q \)[/tex] is [tex]\( \neg q \rightarrow \neg p \)[/tex].
- In words, this means "If the additive inverse is not positive, the number is not negative."
- According to the statement provided, [tex]\( q \)[/tex] is redefined as "a number is negative" and [tex]\( p \)[/tex] as "the additive inverse is positive".
- This means the contrapositive should be formulated in terms of "If the additive inverse is not positive, then the number is not negative."
- Statement: If [tex]\( q = \)[/tex] "a number is negative" and [tex]\( p = \)[/tex] "the additive inverse is positive," the contrapositive of the original statement is [tex]\( \neg p \rightarrow \neg q \)[/tex].
- Evaluation: False
5. Wrong Converse Statement:
- The converse of the statement given is incorrectly framed: "If [tex]\( q = \)[/tex] 'a number is negative' and [tex]\( p = \)[/tex] 'the additive inverse is positive,' the converse of the original statement is [tex]\( q \rightarrow p \)[/tex]."
- However, [tex]\( q \rightarrow p \)[/tex] is correct for converse, but as per the question's presentation, this framing may have an error in syntax.
- Statement: If [tex]\( q = \)[/tex] "a number is negative" and [tex]\( p = \)[/tex] "the additive inverse is positive," the converse of the original statement is [tex]\( q \rightarrow p \)[/tex].
- Evaluation: False due to syntax or misrepresentation problems.
Thus, the true statements are:
1. If [tex]\( p = \)[/tex] "a number is negative" and [tex]\( q = \)[/tex] "the additive inverse is positive," the original statement is [tex]\( p \rightarrow q \)[/tex].
2. If [tex]\( p = \)[/tex] "a number is negative" and [tex]\( q = \)[/tex] "the additive inverse is positive," the inverse of the original statement is [tex]\( \neg p \rightarrow \neg q \)[/tex].
3. If [tex]\( p = \)[/tex] "a number is negative" and [tex]\( q = \)[/tex] "the additive inverse is positive," the converse of the original statement is [tex]\( q \rightarrow p \)[/tex].
True Indices: [1, 2, 3]
