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Write the converse, inverse, and contrapositive of the statement below.
[tex]\[ \sim q \rightarrow \sim r \][/tex]

The converse of the given statement is which of the following?

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]

Sagot :

GinnyAnswer
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]
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