Answer :
Let's analyze the given true statements step-by-step to find the true options:
Original statement: "If a number is negative, the additive inverse is positive."
This means:
- [tex]\( p = \)[/tex] a number is negative
- [tex]\( q = \)[/tex] the additive inverse is positive
1. Original Statement: If [tex]\( p \rightarrow q \)[/tex].
- This translates to: If a number is negative ([tex]\( p \)[/tex]), then the additive inverse is positive ([tex]\( q \)[/tex]).
- True Statement: [tex]\( p \rightarrow q \)[/tex]
2. Inverse of the Original Statement: If [tex]\(\sim p \rightarrow \sim q\)[/tex].
- [tex]\(\sim p =\)[/tex] a number is not negative (a number is non-negative)
- [tex]\(\sim q =\)[/tex] the additive inverse is not positive
- This translates to: If a number is not negative ([tex]\(\sim p \)[/tex]), then the additive inverse is not positive ([tex]\(\sim q \)[/tex]).
- True Statement: [tex]\(\sim p \rightarrow \sim q \)[/tex]
3. Converse of the Original Statement: If [tex]\( q \rightarrow p \)[/tex].
- This translates to: If the additive inverse is positive ([tex]\( q \)[/tex]), then a number is negative ([tex]\( p \)[/tex]).
- True Statement: [tex]\( q \rightarrow p \)[/tex]
4. Contrapositive of the Original Statement: If [tex]\(\sim q \rightarrow \sim p\)[/tex].
- [tex]\(\sim q =\)[/tex] the additive inverse is not positive
- [tex]\(\sim p =\)[/tex] a number is not negative (a number is non-negative)
- This translates to: If the additive inverse is not positive ([tex]\(\sim q \)[/tex]), then a number is not negative ([tex]\(\sim p \)[/tex]).
5. Incorrect Statements:
- "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]\(\sim p \rightarrow \sim q\)[/tex]": This is incorrect because the variables are swapped.
- "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]": This is incorrect because the variables are swapped.
Based on this analysis, the three true options 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]\(\sim p \rightarrow \sim q\)[/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].
Original statement: "If a number is negative, the additive inverse is positive."
This means:
- [tex]\( p = \)[/tex] a number is negative
- [tex]\( q = \)[/tex] the additive inverse is positive
1. Original Statement: If [tex]\( p \rightarrow q \)[/tex].
- This translates to: If a number is negative ([tex]\( p \)[/tex]), then the additive inverse is positive ([tex]\( q \)[/tex]).
- True Statement: [tex]\( p \rightarrow q \)[/tex]
2. Inverse of the Original Statement: If [tex]\(\sim p \rightarrow \sim q\)[/tex].
- [tex]\(\sim p =\)[/tex] a number is not negative (a number is non-negative)
- [tex]\(\sim q =\)[/tex] the additive inverse is not positive
- This translates to: If a number is not negative ([tex]\(\sim p \)[/tex]), then the additive inverse is not positive ([tex]\(\sim q \)[/tex]).
- True Statement: [tex]\(\sim p \rightarrow \sim q \)[/tex]
3. Converse of the Original Statement: If [tex]\( q \rightarrow p \)[/tex].
- This translates to: If the additive inverse is positive ([tex]\( q \)[/tex]), then a number is negative ([tex]\( p \)[/tex]).
- True Statement: [tex]\( q \rightarrow p \)[/tex]
4. Contrapositive of the Original Statement: If [tex]\(\sim q \rightarrow \sim p\)[/tex].
- [tex]\(\sim q =\)[/tex] the additive inverse is not positive
- [tex]\(\sim p =\)[/tex] a number is not negative (a number is non-negative)
- This translates to: If the additive inverse is not positive ([tex]\(\sim q \)[/tex]), then a number is not negative ([tex]\(\sim p \)[/tex]).
5. Incorrect Statements:
- "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]\(\sim p \rightarrow \sim q\)[/tex]": This is incorrect because the variables are swapped.
- "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]": This is incorrect because the variables are swapped.
Based on this analysis, the three true options 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]\(\sim p \rightarrow \sim q\)[/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].
