Answer :
Let's analyze the given statements step-by-step and apply definitions of logical statements.
Given:
- [tex]\( p \)[/tex]: A number is negative
- [tex]\( q \)[/tex]: The additive inverse is positive
The original statement (implication) is [tex]\( p \rightarrow q \)[/tex]:
- If [tex]\( p \)[/tex] (a number is negative), then [tex]\( q \)[/tex] (the additive inverse is positive).
We need to find the corresponding logical variations and assess which, among the provided statements, hold true.
1. Inverse of the Original Statement:
- The inverse of [tex]\( p \rightarrow q \)[/tex] is [tex]\( \sim p \rightarrow \sim q \)[/tex]:
- [tex]\( \sim p \)[/tex]: A number is not negative (non-negative or positive or zero)
- [tex]\( \sim q \)[/tex]: The additive inverse is not positive (non-positive or negative or zero)
- Therefore, the inverse statement is: If a number is not negative, then its additive inverse is not positive.
- Formally, [tex]\( \sim p \rightarrow \sim q \)[/tex].
2. Converse of the Original Statement:
- The converse of [tex]\( p \rightarrow q \)[/tex] is [tex]\( q \rightarrow p \)[/tex]:
- If [tex]\( q \)[/tex] (the additive inverse is positive), then [tex]\( p \)[/tex] (the number is negative).
3. Contrapositive of the Original Statement:
- The contrapositive of [tex]\( p \rightarrow q \)[/tex] is [tex]\( \sim q \rightarrow \sim p \)[/tex]:
- [tex]\( \sim q \)[/tex] (the additive inverse is not positive)
- [tex]\( \sim p \)[/tex] (the number is not negative)
- Therefore, the contrapositive statement is: If the additive inverse is not positive, then the number is not negative.
Given statements:
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].
- True. This matches the original statement directly.
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]\( \sim p \rightarrow \sim q \)[/tex].
- True. This matches what we derived as the inverse.
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]\( \sim q \rightarrow \sim p \)[/tex].
- False. This describes the contrapositive, not the converse. The converse should be [tex]\( q \rightarrow p \)[/tex].
4. 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]\( \sim p \sim \sim q \)[/tex].
- False. This does not match logical definitions. Additionally, the variables [tex]\( p \)[/tex] and [tex]\( q \)[/tex] are interchanged in meaning.
5. 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].
- False. This statement has correct logic but swapped definitions for [tex]\( p \)[/tex] and [tex]\( q \)[/tex].
Based on these evaluations, the three correct statements are:
- 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].
- 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]\( \sim p \rightarrow \sim q \)[/tex].
- 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].
Therefore, the selected three correct options from the problem are correct.
Given:
- [tex]\( p \)[/tex]: A number is negative
- [tex]\( q \)[/tex]: The additive inverse is positive
The original statement (implication) is [tex]\( p \rightarrow q \)[/tex]:
- If [tex]\( p \)[/tex] (a number is negative), then [tex]\( q \)[/tex] (the additive inverse is positive).
We need to find the corresponding logical variations and assess which, among the provided statements, hold true.
1. Inverse of the Original Statement:
- The inverse of [tex]\( p \rightarrow q \)[/tex] is [tex]\( \sim p \rightarrow \sim q \)[/tex]:
- [tex]\( \sim p \)[/tex]: A number is not negative (non-negative or positive or zero)
- [tex]\( \sim q \)[/tex]: The additive inverse is not positive (non-positive or negative or zero)
- Therefore, the inverse statement is: If a number is not negative, then its additive inverse is not positive.
- Formally, [tex]\( \sim p \rightarrow \sim q \)[/tex].
2. Converse of the Original Statement:
- The converse of [tex]\( p \rightarrow q \)[/tex] is [tex]\( q \rightarrow p \)[/tex]:
- If [tex]\( q \)[/tex] (the additive inverse is positive), then [tex]\( p \)[/tex] (the number is negative).
3. Contrapositive of the Original Statement:
- The contrapositive of [tex]\( p \rightarrow q \)[/tex] is [tex]\( \sim q \rightarrow \sim p \)[/tex]:
- [tex]\( \sim q \)[/tex] (the additive inverse is not positive)
- [tex]\( \sim p \)[/tex] (the number is not negative)
- Therefore, the contrapositive statement is: If the additive inverse is not positive, then the number is not negative.
Given statements:
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].
- True. This matches the original statement directly.
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]\( \sim p \rightarrow \sim q \)[/tex].
- True. This matches what we derived as the inverse.
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]\( \sim q \rightarrow \sim p \)[/tex].
- False. This describes the contrapositive, not the converse. The converse should be [tex]\( q \rightarrow p \)[/tex].
4. 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]\( \sim p \sim \sim q \)[/tex].
- False. This does not match logical definitions. Additionally, the variables [tex]\( p \)[/tex] and [tex]\( q \)[/tex] are interchanged in meaning.
5. 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].
- False. This statement has correct logic but swapped definitions for [tex]\( p \)[/tex] and [tex]\( q \)[/tex].
Based on these evaluations, the three correct statements are:
- 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].
- 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]\( \sim p \rightarrow \sim q \)[/tex].
- 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].
Therefore, the selected three correct options from the problem are correct.
