Discover the best answers at Westonci.ca, where experts share their insights and knowledge with you. Discover precise answers to your questions from a wide range of experts on our user-friendly Q&A platform. Get immediate and reliable solutions to your questions from a community of experienced professionals on our platform.
Sagot :
To address the problem, we start by understanding the given statement [tex]\(\sim q \rightarrow \sim r\)[/tex]. Let's break down its logical components and then derive the converse, inverse, and contrapositive.
### Given Statement
[tex]\[ \sim q \rightarrow \sim r \][/tex]
This reads as "if not [tex]\( q \)[/tex], then not [tex]\( r \)[/tex]."
### Converse
The converse of a conditional statement [tex]\(p \rightarrow q\)[/tex] is derived by swapping the hypothesis and conclusion. So, for the given statement [tex]\(\sim q \rightarrow \sim r\)[/tex]:
- Original hypothesis: [tex]\(\sim q\)[/tex]
- Original conclusion: [tex]\(\sim r\)[/tex]
Switch these to get the converse:
[tex]\[ \sim r \rightarrow \sim q \][/tex]
### Answer Choices for the Converse
Given the following options:
A. [tex]\(q \vee \sim r\)[/tex]
B. [tex]\(r \rightarrow q\)[/tex]
C. [tex]\(q \rightarrow r\)[/tex]
D. [tex]\(\sim r \rightarrow \sim q\)[/tex]
The correct choice that matches the derived converse ([tex]\(\sim r \rightarrow \sim q\)[/tex]) is:
[tex]\[ \text{D. } \sim r \rightarrow \sim q \][/tex]
### Inverse
The inverse of a conditional statement [tex]\(p \rightarrow q\)[/tex] is derived by negating both the hypothesis and the conclusion:
- Original statement: [tex]\(\sim q \rightarrow \sim r\)[/tex]
- Negate the hypothesis: [tex]\(q\)[/tex]
- Negate the conclusion: [tex]\(r\)[/tex]
Thus, the inverse is:
[tex]\[ q \rightarrow r \][/tex]
### Contrapositive
The contrapositive of a conditional statement [tex]\(p \rightarrow q\)[/tex] is derived by swapping the hypothesis and conclusion of the inverse.
- Original statement: [tex]\(\sim q \rightarrow \sim r\)[/tex]
- Inverse: [tex]\(q \rightarrow r\)[/tex]
- Swap the hypothesis and conclusion of the inverse:
[tex]\[ \sim r \rightarrow \sim q \][/tex]
Interestingly, the contrapositive and the converse for the given statement are identical in form.
### Summary
- Converse: [tex]\(\sim r \rightarrow \sim q\)[/tex] (Answer: D)
- Inverse: [tex]\(q \rightarrow r\)[/tex]
- Contrapositive: [tex]\(\sim r \rightarrow \sim q\)[/tex]
Therefore, the correct answer to the question "The converse of the given statement is which of the following?" is:
[tex]\[ \text{D. } \sim r \rightarrow \sim q \][/tex]
### Given Statement
[tex]\[ \sim q \rightarrow \sim r \][/tex]
This reads as "if not [tex]\( q \)[/tex], then not [tex]\( r \)[/tex]."
### Converse
The converse of a conditional statement [tex]\(p \rightarrow q\)[/tex] is derived by swapping the hypothesis and conclusion. So, for the given statement [tex]\(\sim q \rightarrow \sim r\)[/tex]:
- Original hypothesis: [tex]\(\sim q\)[/tex]
- Original conclusion: [tex]\(\sim r\)[/tex]
Switch these to get the converse:
[tex]\[ \sim r \rightarrow \sim q \][/tex]
### Answer Choices for the Converse
Given the following options:
A. [tex]\(q \vee \sim r\)[/tex]
B. [tex]\(r \rightarrow q\)[/tex]
C. [tex]\(q \rightarrow r\)[/tex]
D. [tex]\(\sim r \rightarrow \sim q\)[/tex]
The correct choice that matches the derived converse ([tex]\(\sim r \rightarrow \sim q\)[/tex]) is:
[tex]\[ \text{D. } \sim r \rightarrow \sim q \][/tex]
### Inverse
The inverse of a conditional statement [tex]\(p \rightarrow q\)[/tex] is derived by negating both the hypothesis and the conclusion:
- Original statement: [tex]\(\sim q \rightarrow \sim r\)[/tex]
- Negate the hypothesis: [tex]\(q\)[/tex]
- Negate the conclusion: [tex]\(r\)[/tex]
Thus, the inverse is:
[tex]\[ q \rightarrow r \][/tex]
### Contrapositive
The contrapositive of a conditional statement [tex]\(p \rightarrow q\)[/tex] is derived by swapping the hypothesis and conclusion of the inverse.
- Original statement: [tex]\(\sim q \rightarrow \sim r\)[/tex]
- Inverse: [tex]\(q \rightarrow r\)[/tex]
- Swap the hypothesis and conclusion of the inverse:
[tex]\[ \sim r \rightarrow \sim q \][/tex]
Interestingly, the contrapositive and the converse for the given statement are identical in form.
### Summary
- Converse: [tex]\(\sim r \rightarrow \sim q\)[/tex] (Answer: D)
- Inverse: [tex]\(q \rightarrow r\)[/tex]
- Contrapositive: [tex]\(\sim r \rightarrow \sim q\)[/tex]
Therefore, the correct answer to the question "The converse of the given statement is which of the following?" is:
[tex]\[ \text{D. } \sim r \rightarrow \sim q \][/tex]
We hope this information was helpful. Feel free to return anytime for more answers to your questions and concerns. We hope our answers were useful. Return anytime for more information and answers to any other questions you have. Westonci.ca is committed to providing accurate answers. Come back soon for more trustworthy information.