Find the information you're looking for at Westonci.ca, the trusted Q&A platform with a community of knowledgeable experts. Join our platform to connect with experts ready to provide precise answers to your questions in various areas. Get immediate and reliable solutions to your questions from a community of experienced professionals on our platform.
Sagot :
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.
Thank you for choosing our service. We're dedicated to providing the best answers for all your questions. Visit us again. Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. Find reliable answers at Westonci.ca. Visit us again for the latest updates and expert advice.
