To solve this problem, let's carefully break down the steps involving the original conditional statement and its transformations.
### 1. The original statement:
The given conditional statement is:
[tex]\[ p \rightarrow q \][/tex]
### 2. Contrapositive:
The contrapositive of the statement [tex]\( p \rightarrow q \)[/tex] can be formed by negating both the hypothesis and the conclusion and swapping them:
[tex]\[ \sim q \rightarrow \sim p \][/tex]
### 3. Inverse of the Contrapositive:
Next, we need to find the inverse of the contrapositive. The inverse of a statement [tex]\( r \rightarrow s \)[/tex] is found by negating both terms, but not swapping them.
So, for the contrapositive [tex]\( \sim q \rightarrow \sim p \)[/tex], the inverse is:
[tex]\[ \sim(\sim q) \rightarrow \sim(\sim p) \][/tex]
### 4. Simplification:
Now, we simplify the negations:
[tex]\[ \sim(\sim q) \text{ is equivalent to } q \][/tex]
[tex]\[ \sim(\sim p) \text{ is equivalent to } p \][/tex]
Thus, the inverse of the contrapositive [tex]\( \sim q \rightarrow \sim p \)[/tex] simplifies to:
[tex]\[ q \rightarrow p \][/tex]
So, the inverse of the contrapositive of the statement [tex]\( p \rightarrow q \)[/tex] is:
[tex]\[ q \rightarrow p \][/tex]
The correct answer is:
[tex]\[ q \rightarrow p \][/tex]
