To determine which statement is logically equivalent to the conditional statement [tex]\( p \rightarrow q \)[/tex]:
1. Understanding the Conditional Statement:
- A conditional statement [tex]\( p \rightarrow q \)[/tex] reads as "if [tex]\( p \)[/tex] then [tex]\( q \)[/tex]". This means that whenever [tex]\( p \)[/tex] is true, [tex]\( q \)[/tex] must also be true.
2. Identifying Logically Equivalent Statements:
- To find the logically equivalent statement among the given options, we need to consider the contrapositive, inverse, and converse of [tex]\( p \rightarrow q \)[/tex].
- The contrapositive of [tex]\( p \rightarrow q \)[/tex] is [tex]\( \sim q \rightarrow \sim p \)[/tex], which reads as "if [tex]\( q \)[/tex] is not true, then [tex]\( p \)[/tex] is not true". This statement is always logically equivalent to [tex]\( p \rightarrow q \)[/tex].
- The inverse of [tex]\( p \rightarrow q \)[/tex] is [tex]\( \sim p \rightarrow \sim q \)[/tex]. This states that "if [tex]\( p \)[/tex] is not true, then [tex]\( q \)[/tex] is not true". The inverse is not guaranteed to be logically equivalent to [tex]\( p \rightarrow q \)[/tex].
- The converse of [tex]\( p \rightarrow q \)[/tex] is [tex]\( q \rightarrow p \)[/tex]. This states that "if [tex]\( q \)[/tex] is true, then [tex]\( p \)[/tex] is true". The converse is not guaranteed to be logically equivalent to [tex]\( p \rightarrow q \)[/tex].
3. Choosing the Correct Option:
- From the definitions above, the contrapositive [tex]\( \sim q \rightarrow \sim p \)[/tex] is logically equivalent to [tex]\( p \rightarrow q \)[/tex].
Considering the given options:
- [tex]\( \sim p \rightarrow \sim q \)[/tex] (Inverse) - Not equivalent.
- [tex]\( \sim q \rightarrow \sim p \)[/tex] (Contrapositive) - Equivalent.
- [tex]\( q \rightarrow p \)[/tex] (Converse) - Not equivalent.
- [tex]\( p \rightarrow \sim q \)[/tex] - Not equivalent and a different logical meaning.
Thus, the logically equivalent statement to [tex]\( p \rightarrow q \)[/tex] is:
[tex]\[
\boxed{\sim q \rightarrow \sim p}
\][/tex]
