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]
