To determine which statement [tex]\( q \rightarrow p \)[/tex] represents, we need to understand the different forms of conditional statements involving [tex]\( p \)[/tex] and [tex]\( q \)[/tex]. Here is a detailed explanation:
1. Original Conditional Statement: The original conditional statement is formulated as [tex]\( p \rightarrow q \)[/tex]. This means "If [tex]\( p \)[/tex], then [tex]\( q \)[/tex]."
2. Inverse of the Original Conditional Statement: The inverse negates both the hypothesis and the conclusion of the original conditional statement, represented as [tex]\( \neg p \rightarrow \neg q \)[/tex]. This means "If not [tex]\( p \)[/tex], then not [tex]\( q \)[/tex]."
3. Converse of the Original Conditional Statement: The converse switches the hypothesis and the conclusion of the original conditional statement, represented as [tex]\( q \rightarrow p \)[/tex]. This means "If [tex]\( q \)[/tex], then [tex]\( p \)[/tex]."
4. Contrapositive of the Original Conditional Statement: The contrapositive switches and negates both the hypothesis and the conclusion of the original conditional statement, represented as [tex]\( \neg q \rightarrow \neg p \)[/tex]. This means "If not [tex]\( q \)[/tex], then not [tex]\( p \)[/tex]."
Given [tex]\( q \rightarrow p \)[/tex]:
- It switches the hypothesis [tex]\( p \)[/tex] and the conclusion [tex]\( q \)[/tex] of the original conditional statement [tex]\( p \rightarrow q \)[/tex].
Therefore, [tex]\( q \rightarrow p \)[/tex] is the converse of [tex]\( p \rightarrow q \)[/tex].
Answer: the converse of the original conditional statement
