To solve the problem, we need to understand the different forms of a conditional statement [tex]\( p \rightarrow q \)[/tex]. Here are the definitions:
1. Original conditional statement: [tex]\( p \rightarrow q \)[/tex]
- This states that if [tex]\( p \)[/tex] is true, then [tex]\( q \)[/tex] must also be true.
2. Converse of the original conditional statement: [tex]\( q \rightarrow p \)[/tex]
- This is formed by swapping the hypothesis and the conclusion of the original statement. If [tex]\( q \)[/tex] is true, then [tex]\( p \)[/tex] must also be true.
3. Contrapositive of the original conditional statement: [tex]\( \sim q \rightarrow \sim p \)[/tex]
- The contrapositive is formed by negating both the hypothesis and conclusion of the original statement and then reversing them. If [tex]\( q \)[/tex] is not true, then [tex]\( p \)[/tex] is not true either.
4. Inverse of the original conditional statement: [tex]\( \sim p \rightarrow \sim q \)[/tex]
- The inverse is formed by negating both the hypothesis and conclusion of the original statement. If [tex]\( p \)[/tex] is not true, then [tex]\( q \)[/tex] is not true either.
Given the statement [tex]\( \sim p \rightarrow \sim q \)[/tex]:
- This statement negates both the hypothesis ([tex]\( \sim p \)[/tex]) and conclusion ([tex]\( \sim q \)[/tex]) of the original statement [tex]\( p \rightarrow q \)[/tex].
From the definitions provided above, this corresponds to the inverse of the original conditional statement.
Thus, [tex]\(\sim p \rightarrow \sim q\)[/tex] represents the inverse of the original conditional statement.
Therefore, the answer is:
the inverse of the original conditional statement.
