Answer :
To solve this problem, let’s break it down into steps and analyze each logical statement given.
### 1. Original Statement:
The original statement is [tex]\( p \rightarrow q \)[/tex], where:
- [tex]\( p = \text{"a number is negative"} \)[/tex]
- [tex]\( q = \text{"the additive inverse is positive"} \)[/tex]
So, the original statement reads:
- "If a number is negative, the additive inverse is positive."
### 2. Inverse of the Original Statement:
The inverse of the original statement negates both [tex]\( p \)[/tex] and [tex]\( q \)[/tex] and keeps their order, which is [tex]\( \sim p \rightarrow \sim q \)[/tex].
- [tex]\( \sim p = \text{"a number is not negative"} \)[/tex]
- [tex]\( \sim q = \text{"the additive inverse is not positive"} \)[/tex]
So, the inverse statement is:
- "If a number is not negative, the additive inverse is not positive."
### 3. Converse of the Original Statement:
The converse statement switches [tex]\( p \)[/tex] and [tex]\( q \)[/tex], which is [tex]\( q \rightarrow p \)[/tex].
If we switch [tex]\( p \)[/tex] and [tex]\( q \)[/tex] according to the values of [tex]\( q \)[/tex] and [tex]\( p \)[/tex]:
- [tex]\( q = \text{"the additive inverse is positive"} \)[/tex]
- [tex]\( p = \text{"a number is negative"} \)[/tex]
So, the converse statement is:
- "If the additive inverse is positive, the number is negative."
### 4. Contrapositive of the Original Statement:
The contrapositive of the original statement negates and swaps [tex]\( p \)[/tex] and [tex]\( q \)[/tex], which is [tex]\( \sim q \rightarrow \sim p \)[/tex].
- [tex]\( \sim q = \text{"the additive inverse is not positive"} \)[/tex]
- [tex]\( \sim p = \text{"a number is not negative"} \)[/tex]
So, the contrapositive statement is:
- "If the additive inverse is not positive, the number is not negative."
### 5. Converse of the Original Statement with Redefined [tex]\( p \)[/tex] and [tex]\( q \)[/tex]:
If we take the swapped [tex]\( p \)[/tex] and [tex]\( q \)[/tex] as given:
- New [tex]\( p = \text{"the additive inverse is positive"} \)[/tex]
- New [tex]\( q = \text{"a number is negative"} \)[/tex]
Then, the converse statement as [tex]\( p \rightarrow q \)[/tex] becomes:
- "If the additive inverse is positive, a number is negative."
### Reviewing the options:
1. The original statement is: "If a number is negative, the additive inverse is positive." [tex]\( p \rightarrow q \)[/tex]
2. The inverse of the original statement is: "If a number is not negative, the additive inverse is not positive." [tex]\( \sim p \rightarrow \sim q \)[/tex]
3. The converse of the original statement is: "If the additive inverse is positive, the number is negative." [tex]\( q \rightarrow p \)[/tex]
4. The contrapositive of the original statement is: "If the additive inverse is not positive, the number is not negative." [tex]\( \sim q \rightarrow \sim p \)[/tex]
5. The converse of the original statement with redefined [tex]\( p \)[/tex] and [tex]\( q \)[/tex] is: "If the additive inverse is positive, a number is negative." [tex]\( q \rightarrow p \)[/tex]
Based on the information provided, the true statements are options 1, 2, and 3.
So, the selected options that are true are:
- The original statement.
- The inverse of the original statement.
- The converse of the original statement.
### 1. Original Statement:
The original statement is [tex]\( p \rightarrow q \)[/tex], where:
- [tex]\( p = \text{"a number is negative"} \)[/tex]
- [tex]\( q = \text{"the additive inverse is positive"} \)[/tex]
So, the original statement reads:
- "If a number is negative, the additive inverse is positive."
### 2. Inverse of the Original Statement:
The inverse of the original statement negates both [tex]\( p \)[/tex] and [tex]\( q \)[/tex] and keeps their order, which is [tex]\( \sim p \rightarrow \sim q \)[/tex].
- [tex]\( \sim p = \text{"a number is not negative"} \)[/tex]
- [tex]\( \sim q = \text{"the additive inverse is not positive"} \)[/tex]
So, the inverse statement is:
- "If a number is not negative, the additive inverse is not positive."
### 3. Converse of the Original Statement:
The converse statement switches [tex]\( p \)[/tex] and [tex]\( q \)[/tex], which is [tex]\( q \rightarrow p \)[/tex].
If we switch [tex]\( p \)[/tex] and [tex]\( q \)[/tex] according to the values of [tex]\( q \)[/tex] and [tex]\( p \)[/tex]:
- [tex]\( q = \text{"the additive inverse is positive"} \)[/tex]
- [tex]\( p = \text{"a number is negative"} \)[/tex]
So, the converse statement is:
- "If the additive inverse is positive, the number is negative."
### 4. Contrapositive of the Original Statement:
The contrapositive of the original statement negates and swaps [tex]\( p \)[/tex] and [tex]\( q \)[/tex], which is [tex]\( \sim q \rightarrow \sim p \)[/tex].
- [tex]\( \sim q = \text{"the additive inverse is not positive"} \)[/tex]
- [tex]\( \sim p = \text{"a number is not negative"} \)[/tex]
So, the contrapositive statement is:
- "If the additive inverse is not positive, the number is not negative."
### 5. Converse of the Original Statement with Redefined [tex]\( p \)[/tex] and [tex]\( q \)[/tex]:
If we take the swapped [tex]\( p \)[/tex] and [tex]\( q \)[/tex] as given:
- New [tex]\( p = \text{"the additive inverse is positive"} \)[/tex]
- New [tex]\( q = \text{"a number is negative"} \)[/tex]
Then, the converse statement as [tex]\( p \rightarrow q \)[/tex] becomes:
- "If the additive inverse is positive, a number is negative."
### Reviewing the options:
1. The original statement is: "If a number is negative, the additive inverse is positive." [tex]\( p \rightarrow q \)[/tex]
2. The inverse of the original statement is: "If a number is not negative, the additive inverse is not positive." [tex]\( \sim p \rightarrow \sim q \)[/tex]
3. The converse of the original statement is: "If the additive inverse is positive, the number is negative." [tex]\( q \rightarrow p \)[/tex]
4. The contrapositive of the original statement is: "If the additive inverse is not positive, the number is not negative." [tex]\( \sim q \rightarrow \sim p \)[/tex]
5. The converse of the original statement with redefined [tex]\( p \)[/tex] and [tex]\( q \)[/tex] is: "If the additive inverse is positive, a number is negative." [tex]\( q \rightarrow p \)[/tex]
Based on the information provided, the true statements are options 1, 2, and 3.
So, the selected options that are true are:
- The original statement.
- The inverse of the original statement.
- The converse of the original statement.
