To determine which statement is logically equivalent to a given conditional statement [tex]\( p \rightarrow q \)[/tex], we need to consider a few key concepts in logic: the contrapositive, the inverse, and the converse of a conditional statement.
1. Original Statement: [tex]\( p \rightarrow q \)[/tex]
- This translates to: "If [tex]\( p \)[/tex] is true, then [tex]\( q \)[/tex] is true".
2. Contrapositive: [tex]\( \sim q \rightarrow \sim p \)[/tex]
- This translates to: "If [tex]\( q \)[/tex] is not true, then [tex]\( p \)[/tex] is not true".
- A fundamental property in logic is that the contrapositive of a conditional statement is always logically equivalent to the original conditional statement.
3. Inverse: [tex]\( \sim p \rightarrow \sim q \)[/tex]
- This translates to: "If [tex]\( p \)[/tex] is not true, then [tex]\( q \)[/tex] is not true".
- The inverse is not logically equivalent to the original statement.
4. Converse: [tex]\( q \rightarrow p \)[/tex]
- This translates to: "If [tex]\( q \)[/tex] is true, then [tex]\( p \)[/tex] is true".
- The converse is not logically equivalent to the original statement either.
5. Other Option: [tex]\( p \rightarrow \sim q \)[/tex]
- This translates to: "If [tex]\( p \)[/tex] is true, then [tex]\( q \)[/tex] is not true".
- This statement changes the relationship between [tex]\( p \)[/tex] and [tex]\( q \)[/tex] and is not logically equivalent to the original statement.
Given these concepts, the statement that is logically equivalent to [tex]\( p \rightarrow q \)[/tex] is the contrapositive: [tex]\( \sim q \rightarrow \sim p \)[/tex].
Therefore, the correct choice is:
[tex]\[ \sim q \rightarrow \sim p \][/tex]
This corresponds to the second option from the given choices:
[tex]\[ \boxed{\sim q \rightarrow \sim p} \][/tex]
