Answered

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Question 8 of 10

Write the converse, inverse, and contrapositive of the statement below:

[tex]\[ \sim q \rightarrow \sim r \][/tex]

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

A. [tex]\[ r \rightarrow q \][/tex]

B. [tex]\[ \sim r \rightarrow \sim q \][/tex]

C. [tex]\[ q \rightarrow r \][/tex]

D. [tex]\[ q \vee \sim r \][/tex]

Sagot :

GinnyAnswer
Certainly! To solve this problem, we need to determine the inverse of the given logical statement. Here’s a detailed, step-by-step breakdown:

Given statement:
[tex]\[ \sim q \rightarrow \sim r \][/tex]
This statement reads as "if not [tex]\( q \)[/tex], then not [tex]\( r \)[/tex]".

### Step-by-Step Process to Find the Inverse:

1. Understand the Components of the Statement:
- The hypothesis (or antecedent) is [tex]\( \sim q \)[/tex].
- The conclusion (or consequent) is [tex]\( \sim r \)[/tex].

2. Formulate the Inverse:
- The inverse of a conditional statement [tex]\( p \rightarrow q \)[/tex] is formed by negating both the hypothesis and the conclusion.
- Thus, the inverse of [tex]\( \sim q \rightarrow \sim r \)[/tex] requires us to negate [tex]\( \sim q \)[/tex] and [tex]\( \sim r \)[/tex]:
- Negate [tex]\( \sim q \)[/tex], which gives us [tex]\( q \)[/tex].
- Negate [tex]\( \sim r \)[/tex], which gives us [tex]\( r \)[/tex].

3. Construct the Inverse Statement:
- The inverse of [tex]\( \sim q \rightarrow \sim r \)[/tex] is therefore [tex]\( q \rightarrow r \)[/tex].
- This reads as "if [tex]\( q \)[/tex], then [tex]\( r \)[/tex]".

### Finalize the Answer:
Now, we need to match our result with the given options:

A. [tex]\( r \rightarrow q \)[/tex]
B. [tex]\( \sim r \rightarrow \sim q \)[/tex]
C. [tex]\( q \rightarrow r \)[/tex]
D. [tex]\( q \lor \sim r \)[/tex]

From the constructed inverse statement, we see that the correct answer is:

C. [tex]\( q \rightarrow r \)[/tex]
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