To determine which statement is logically equivalent to the given conditional statement [tex]\( \sim p \rightarrow q \)[/tex], let's analyze it in detail.
The statement [tex]\( \sim p \rightarrow q \)[/tex] means "if not [tex]\( p \)[/tex], then [tex]\( q \)[/tex]." To find a logically equivalent statement, we often use the concept of contrapositive in logic. The contrapositive of an implication [tex]\( A \rightarrow B \)[/tex] is [tex]\( \sim B \rightarrow \sim A \)[/tex]. The original statement and its contrapositive are always logically equivalent.
Given our statement [tex]\( \sim p \rightarrow q \)[/tex], let's identify its contrapositive:
1. The original statement is [tex]\( \sim p \rightarrow q \)[/tex].
2. To form the contrapositive, we switch the hypothesis and conclusion and negate both. This gives us [tex]\( \sim q \)[/tex] as the new hypothesis and [tex]\( \sim (\sim p) \)[/tex] as the new conclusion.
3. Negating [tex]\( \sim p \)[/tex] results in [tex]\( p \)[/tex]. Therefore, the contrapositive of [tex]\( \sim p \rightarrow q \)[/tex] is [tex]\( \sim q \rightarrow p \)[/tex].
Therefore, the statement that is logically equivalent to [tex]\( \sim p \rightarrow q \)[/tex] is:
[tex]\[ \sim q \rightarrow p \][/tex]
So, the correct answer is:
[tex]\[ \sim q \rightarrow p \][/tex]
Hence, the equivalent statement is the fourth option provided:
[tex]\[ \sim q \rightarrow p \][/tex]
