Answer :
To solve this problem, we need to understand the nature of different forms of conditional statements in logic. Let's start by defining these terms:
1. Original Conditional Statement: This is typically written as [tex]\( p \rightarrow q \)[/tex], where [tex]\( p \)[/tex] is the hypothesis and [tex]\( q \)[/tex] is the conclusion. It reads as "if [tex]\( p \)[/tex], then [tex]\( q \)[/tex]".
2. Converse: This statement reverses the hypothesis and the conclusion. The converse of [tex]\( p \rightarrow q \)[/tex] is [tex]\( q \rightarrow p \)[/tex]. It reads as "if [tex]\( q \)[/tex], then [tex]\( p \)[/tex]".
3. Inverse: This statement negates both the hypothesis and the conclusion of the original statement. The inverse of [tex]\( p \rightarrow q \)[/tex] is [tex]\( \neg p \rightarrow \neg q \)[/tex]. It reads as "if not [tex]\( p \)[/tex], then not [tex]\( q \)[/tex]".
4. Contrapositive: This statement both reverses and negates the original hypothesis and conclusion. The contrapositive of [tex]\( p \rightarrow q \)[/tex] is [tex]\( \neg q \rightarrow \neg p \)[/tex]. It reads as "if not [tex]\( q \)[/tex], then not [tex]\( p \)[/tex]".
Given the statement [tex]\( q \rightarrow p \)[/tex], we notice that the hypothesis ([tex]\( p \)[/tex]) and conclusion ([tex]\( q \)[/tex]) have been reversed compared to the original conditional statement [tex]\( p \rightarrow q \)[/tex].
Thus, [tex]\( q \rightarrow p \)[/tex] represents the converse of the original conditional statement [tex]\( p \rightarrow q \)[/tex].
Therefore, the correct answer is:
- The converse of the original conditional statement
So, [tex]\( q \rightarrow p \)[/tex] is the converse of the original conditional statement.
1. Original Conditional Statement: This is typically written as [tex]\( p \rightarrow q \)[/tex], where [tex]\( p \)[/tex] is the hypothesis and [tex]\( q \)[/tex] is the conclusion. It reads as "if [tex]\( p \)[/tex], then [tex]\( q \)[/tex]".
2. Converse: This statement reverses the hypothesis and the conclusion. The converse of [tex]\( p \rightarrow q \)[/tex] is [tex]\( q \rightarrow p \)[/tex]. It reads as "if [tex]\( q \)[/tex], then [tex]\( p \)[/tex]".
3. Inverse: This statement negates both the hypothesis and the conclusion of the original statement. The inverse of [tex]\( p \rightarrow q \)[/tex] is [tex]\( \neg p \rightarrow \neg q \)[/tex]. It reads as "if not [tex]\( p \)[/tex], then not [tex]\( q \)[/tex]".
4. Contrapositive: This statement both reverses and negates the original hypothesis and conclusion. The contrapositive of [tex]\( p \rightarrow q \)[/tex] is [tex]\( \neg q \rightarrow \neg p \)[/tex]. It reads as "if not [tex]\( q \)[/tex], then not [tex]\( p \)[/tex]".
Given the statement [tex]\( q \rightarrow p \)[/tex], we notice that the hypothesis ([tex]\( p \)[/tex]) and conclusion ([tex]\( q \)[/tex]) have been reversed compared to the original conditional statement [tex]\( p \rightarrow q \)[/tex].
Thus, [tex]\( q \rightarrow p \)[/tex] represents the converse of the original conditional statement [tex]\( p \rightarrow q \)[/tex].
Therefore, the correct answer is:
- The converse of the original conditional statement
So, [tex]\( q \rightarrow p \)[/tex] is the converse of the original conditional statement.
