To convert a conditional statement into its converse, inverse, or contrapositive, one must understand what each term represents. Let's break down each possibility given the original statement [tex]\( p \rightarrow q \)[/tex]:
1. Original Conditional Statement ([tex]\( p \rightarrow q \)[/tex]): This is the given statement where [tex]\( p \)[/tex] is the hypothesis and [tex]\( q \)[/tex] is the conclusion.
2. Inverse ([tex]\( \neg p \rightarrow \neg q \)[/tex]): The inverse of the original statement negates both the hypothesis and the conclusion. So, if the original statement is [tex]\( p \rightarrow q \)[/tex], the inverse is [tex]\( \neg p \rightarrow \neg q \)[/tex].
3. Converse ([tex]\( q \rightarrow p \)[/tex]): The converse of the original statement switches the hypothesis and the conclusion. So, if the original statement is [tex]\( p \rightarrow q \)[/tex], the converse is [tex]\( q \rightarrow p \)[/tex].
4. Contrapositive ([tex]\( \neg q \rightarrow \neg p \)[/tex]): The contrapositive of the original statement negates and switches both the hypothesis and the conclusion. So, if the original statement is [tex]\( p \rightarrow q \)[/tex], the contrapositive is [tex]\( \neg q \rightarrow \neg p \)[/tex].
Given the original statement [tex]\( p \rightarrow q \)[/tex], if we switch the hypothesis and the conclusion, we get [tex]\( q \rightarrow p \)[/tex].
Thus, [tex]\( q \rightarrow p \)[/tex] is the converse of the original conditional statement.
